id	sid	tid	token	lemma	pos
ejpam-6813	1	1	european	european	PROPN
ejpam-6813	1	2	journal	journal	PROPN
ejpam-6813	1	3	of	of	ADP
ejpam-6813	1	4	pure	pure	ADJ
ejpam-6813	1	5	and	and	CCONJ
ejpam-6813	1	6	applied	applied	ADJ
ejpam-6813	1	7	mathematics	mathematic	NOUN
ejpam-6813	1	8	2025	2025	NUM
ejpam-6813	1	9	,	,	PUNCT
ejpam-6813	1	10	vol	vol	NOUN
ejpam-6813	1	11	.	.	PROPN
ejpam-6813	1	12	18	18	NUM
ejpam-6813	1	13	,	,	PUNCT
ejpam-6813	1	14	issue	issue	NOUN
ejpam-6813	1	15	4	4	NUM
ejpam-6813	1	16	,	,	PUNCT
ejpam-6813	1	17	article	article	NOUN
ejpam-6813	1	18	number	number	NOUN
ejpam-6813	1	19	6813	6813	NUM
ejpam-6813	1	20	issn	issn	PROPN
ejpam-6813	1	21	1307	1307	NUM
ejpam-6813	1	22	-	-	SYM
ejpam-6813	1	23	5543	5543	NUM
ejpam-6813	1	24	–	–	PUNCT
ejpam-6813	1	25	ejpam.com	ejpam.com	X
ejpam-6813	1	26	published	publish	VERB
ejpam-6813	1	27	by	by	ADP
ejpam-6813	1	28	new	new	PROPN
ejpam-6813	1	29	york	york	PROPN
ejpam-6813	1	30	business	business	PROPN
ejpam-6813	1	31	global	global	ADJ
ejpam-6813	1	32	properties	property	NOUN
ejpam-6813	1	33	and	and	CCONJ
ejpam-6813	1	34	applications	application	NOUN
ejpam-6813	1	35	of	of	ADP
ejpam-6813	1	36	generalized	generalized	ADJ
ejpam-6813	1	37	numerical	numerical	ADJ
ejpam-6813	1	38	radius	radius	NOUN
ejpam-6813	1	39	in	in	ADP
ejpam-6813	1	40	block	block	NOUN
ejpam-6813	1	41	matrix	matrix	NOUN
ejpam-6813	1	42	structures	structure	NOUN
ejpam-6813	1	43	raja’a	raja’a	VERB
ejpam-6813	1	44	al	al	PROPN
ejpam-6813	1	45	-	-	PUNCT
ejpam-6813	1	46	naimi1,2,1	naimi1,2,1	PROPN
ejpam-6813	1	47	,	,	PUNCT
ejpam-6813	1	48	manal	manal	PROPN
ejpam-6813	1	49	al	al	PROPN
ejpam-6813	1	50	-	-	PUNCT
ejpam-6813	1	51	labadi2	labadi2	PROPN
ejpam-6813	1	52	,	,	PUNCT
ejpam-6813	1	53	wasim	wasim	PROPN
ejpam-6813	1	54	audeh2	audeh2	PROPN
ejpam-6813	1	55	,	,	PUNCT
ejpam-6813	1	56	jamal	jamal	PROPN
ejpam-6813	1	57	oudetallah2	oudetallah2	PROPN
ejpam-6813	1	58	,	,	PUNCT
ejpam-6813	1	59	mutti	mutti	PROPN
ejpam-6813	1	60	-	-	PUNCT
ejpam-6813	1	61	ur	ur	PROPN
ejpam-6813	1	62	rehman3	rehman3	PROPN
ejpam-6813	1	63	,	,	PUNCT
ejpam-6813	1	64	dheyaa	dheyaa	PROPN
ejpam-6813	1	65	alangood4	alangood4	PROPN
ejpam-6813	1	66	1	1	NUM
ejpam-6813	1	67	department	department	NOUN
ejpam-6813	1	68	of	of	ADP
ejpam-6813	1	69	mathematics	mathematic	NOUN
ejpam-6813	1	70	,	,	PUNCT
ejpam-6813	1	71	faculty	faculty	NOUN
ejpam-6813	1	72	of	of	ADP
ejpam-6813	1	73	mathematics	mathematic	NOUN
ejpam-6813	1	74	and	and	CCONJ
ejpam-6813	1	75	data	datum	NOUN
ejpam-6813	1	76	science	science	NOUN
ejpam-6813	1	77	,	,	PUNCT
ejpam-6813	1	78	emirates	emirates	PROPN
ejpam-6813	1	79	aviation	aviation	PROPN
ejpam-6813	1	80	university	university	PROPN
ejpam-6813	1	81	,	,	PUNCT
ejpam-6813	1	82	united	united	PROPN
ejpam-6813	1	83	arab	arab	PROPN
ejpam-6813	1	84	emirates	emirates	PROPN
ejpam-6813	1	85	2	2	NUM
ejpam-6813	1	86	department	department	NOUN
ejpam-6813	1	87	of	of	ADP
ejpam-6813	1	88	mathematics	mathematic	NOUN
ejpam-6813	1	89	,	,	PUNCT
ejpam-6813	1	90	faculty	faculty	NOUN
ejpam-6813	1	91	of	of	ADP
ejpam-6813	1	92	arts	art	NOUN
ejpam-6813	1	93	and	and	CCONJ
ejpam-6813	1	94	sciences	science	NOUN
ejpam-6813	1	95	,	,	PUNCT
ejpam-6813	1	96	university	university	NOUN
ejpam-6813	1	97	of	of	ADP
ejpam-6813	1	98	petra	petra	PROPN
ejpam-6813	1	99	,	,	PUNCT
ejpam-6813	1	100	amman	amman	PROPN
ejpam-6813	1	101	,	,	PUNCT
ejpam-6813	1	102	jordan	jordan	PROPN
ejpam-6813	1	103	3	3	NUM
ejpam-6813	1	104	center	center	NOUN
ejpam-6813	1	105	of	of	ADP
ejpam-6813	1	106	research	research	NOUN
ejpam-6813	1	107	and	and	CCONJ
ejpam-6813	1	108	innovation	innovation	NOUN
ejpam-6813	1	109	,	,	PUNCT
ejpam-6813	1	110	asia	asia	PROPN
ejpam-6813	1	111	international	international	PROPN
ejpam-6813	1	112	university	university	PROPN
ejpam-6813	1	113	,	,	PUNCT
ejpam-6813	1	114	200100	200100	NUM
ejpam-6813	1	115	,	,	PUNCT
ejpam-6813	1	116	bukhara	bukhara	PROPN
ejpam-6813	1	117	,	,	PUNCT
ejpam-6813	1	118	uzbekistan	uzbekistan	PROPN
ejpam-6813	1	119	4	4	NUM
ejpam-6813	1	120	university	university	NOUN
ejpam-6813	1	121	of	of	ADP
ejpam-6813	1	122	tikrit	tikrit	NOUN
ejpam-6813	1	123	,	,	PUNCT
ejpam-6813	1	124	college	college	NOUN
ejpam-6813	1	125	of	of	ADP
ejpam-6813	1	126	basic	basic	ADJ
ejpam-6813	1	127	education	education	NOUN
ejpam-6813	1	128	,	,	PUNCT
ejpam-6813	1	129	shirqat	shirqat	PROPN
ejpam-6813	1	130	,	,	PUNCT
ejpam-6813	1	131	iraq	iraq	PROPN
ejpam-6813	1	132	abstract	abstract	NOUN
ejpam-6813	1	133	.	.	PUNCT
ejpam-6813	2	1	in	in	ADP
ejpam-6813	2	2	this	this	DET
ejpam-6813	2	3	paper	paper	NOUN
ejpam-6813	2	4	,	,	PUNCT
ejpam-6813	2	5	we	we	PRON
ejpam-6813	2	6	prove	prove	VERB
ejpam-6813	2	7	several	several	ADJ
ejpam-6813	2	8	results	result	NOUN
ejpam-6813	2	9	that	that	PRON
ejpam-6813	2	10	generalize	generalize	VERB
ejpam-6813	2	11	fundamental	fundamental	ADJ
ejpam-6813	2	12	properties	property	NOUN
ejpam-6813	2	13	of	of	ADP
ejpam-6813	2	14	numerical	numerical	ADJ
ejpam-6813	2	15	radius	radius	PROPN
ejpam-6813	2	16	and	and	CCONJ
ejpam-6813	2	17	generalized	generalized	ADJ
ejpam-6813	2	18	numerical	numerical	ADJ
ejpam-6813	2	19	radius	radius	NOUN
ejpam-6813	2	20	.	.	PUNCT
ejpam-6813	3	1	among	among	ADP
ejpam-6813	3	2	our	our	PRON
ejpam-6813	3	3	proven	prove	VERB
ejpam-6813	3	4	inequalities	inequality	NOUN
ejpam-6813	3	5	,	,	PUNCT
ejpam-6813	3	6	we	we	PRON
ejpam-6813	3	7	establish	establish	VERB
ejpam-6813	3	8	theorems	theorem	NOUN
ejpam-6813	3	9	for	for	ADP
ejpam-6813	3	10	matrices	matrix	NOUN
ejpam-6813	3	11	a	a	DET
ejpam-6813	3	12	∈	∈	PROPN
ejpam-6813	3	13	mm(mn	mm(mn	NOUN
ejpam-6813	3	14	)	)	PUNCT
ejpam-6813	3	15	,	,	PUNCT
ejpam-6813	3	16	where	where	SCONJ
ejpam-6813	3	17	mm(mn	mm(mn	NOUN
ejpam-6813	3	18	)	)	PUNCT
ejpam-6813	3	19	represents	represent	VERB
ejpam-6813	3	20	the	the	DET
ejpam-6813	3	21	set	set	NOUN
ejpam-6813	3	22	of	of	ADP
ejpam-6813	3	23	all	all	DET
ejpam-6813	3	24	m	m	NOUN
ejpam-6813	3	25	×	×	ADJ
ejpam-6813	3	26	m	m	NOUN
ejpam-6813	3	27	block	block	NOUN
ejpam-6813	3	28	complex	complex	ADJ
ejpam-6813	3	29	matrices	matrix	NOUN
ejpam-6813	3	30	with	with	ADP
ejpam-6813	3	31	each	each	DET
ejpam-6813	3	32	block	block	NOUN
ejpam-6813	3	33	belonging	belong	VERB
ejpam-6813	3	34	to	to	ADP
ejpam-6813	3	35	mn(c	mn(c	NOUN
ejpam-6813	3	36	)	)	PUNCT
ejpam-6813	3	37	.	.	PUNCT
ejpam-6813	4	1	furthermore	furthermore	ADV
ejpam-6813	4	2	,	,	PUNCT
ejpam-6813	4	3	we	we	PRON
ejpam-6813	4	4	demonstrate	demonstrate	VERB
ejpam-6813	4	5	that	that	SCONJ
ejpam-6813	4	6	if	if	SCONJ
ejpam-6813	4	7	a	a	DET
ejpam-6813	4	8	∈	∈	PROPN
ejpam-6813	4	9	m2(mn	m2(mn	PROPN
ejpam-6813	4	10	)	)	PUNCT
ejpam-6813	4	11	is	be	AUX
ejpam-6813	4	12	a	a	DET
ejpam-6813	4	13	positive	positive	ADJ
ejpam-6813	4	14	semidefinite	semidefinite	NOUN
ejpam-6813	4	15	matrix	matrix	NOUN
ejpam-6813	4	16	,	,	PUNCT
ejpam-6813	4	17	then	then	ADV
ejpam-6813	4	18	w(2)(a	w(2)(a	NOUN
ejpam-6813	4	19	)	)	PUNCT
ejpam-6813	4	20	is	be	AUX
ejpam-6813	4	21	positive	positive	ADJ
ejpam-6813	4	22	semidefinite	semidefinite	NOUN
ejpam-6813	4	23	,	,	PUNCT
ejpam-6813	4	24	and	and	CCONJ
ejpam-6813	4	25	if	if	SCONJ
ejpam-6813	4	26	a	a	DET
ejpam-6813	4	27	∈	∈	PROPN
ejpam-6813	4	28	mm(m2	mm(m2	NOUN
ejpam-6813	4	29	)	)	PUNCT
ejpam-6813	4	30	is	be	AUX
ejpam-6813	4	31	a	a	DET
ejpam-6813	4	32	positive	positive	ADJ
ejpam-6813	4	33	semidefinite	semidefinite	NOUN
ejpam-6813	4	34	matrix	matrix	NOUN
ejpam-6813	4	35	,	,	PUNCT
ejpam-6813	4	36	then	then	ADV
ejpam-6813	4	37	w(1)(a	w(1)(a	NOUN
ejpam-6813	4	38	)	)	PUNCT
ejpam-6813	4	39	is	be	AUX
ejpam-6813	4	40	positive	positive	ADJ
ejpam-6813	4	41	semidefinite	semidefinite	NOUN
ejpam-6813	4	42	.	.	PUNCT
ejpam-6813	5	1	here	here	ADV
ejpam-6813	5	2	,	,	PUNCT
ejpam-6813	5	3	w(1)(a	w(1)(a	NOUN
ejpam-6813	5	4	)	)	PUNCT
ejpam-6813	5	5	denotes	denote	VERB
ejpam-6813	5	6	the	the	DET
ejpam-6813	5	7	first	first	ADJ
ejpam-6813	5	8	partial	partial	ADJ
ejpam-6813	5	9	matrix	matrix	NOUN
ejpam-6813	5	10	of	of	ADP
ejpam-6813	5	11	numerical	numerical	ADJ
ejpam-6813	5	12	radius	radius	NOUN
ejpam-6813	5	13	and	and	CCONJ
ejpam-6813	5	14	w(2)(a	w(2)(a	NOUN
ejpam-6813	5	15	)	)	PUNCT
ejpam-6813	5	16	denotes	denote	VERB
ejpam-6813	5	17	the	the	DET
ejpam-6813	5	18	second	second	ADJ
ejpam-6813	5	19	partial	partial	ADJ
ejpam-6813	5	20	matrix	matrix	NOUN
ejpam-6813	5	21	of	of	ADP
ejpam-6813	5	22	numerical	numerical	ADJ
ejpam-6813	5	23	radius	radius	NOUN
ejpam-6813	5	24	,	,	PUNCT
ejpam-6813	5	25	respectively	respectively	ADV
ejpam-6813	5	26	.	.	PUNCT
ejpam-6813	6	1	additionally	additionally	ADV
ejpam-6813	6	2	,	,	PUNCT
ejpam-6813	6	3	we	we	PRON
ejpam-6813	6	4	develop	develop	VERB
ejpam-6813	6	5	relationships	relationship	NOUN
ejpam-6813	6	6	with	with	ADP
ejpam-6813	6	7	classical	classical	ADJ
ejpam-6813	6	8	matrix	matrix	NOUN
ejpam-6813	6	9	parameters	parameter	NOUN
ejpam-6813	6	10	and	and	CCONJ
ejpam-6813	6	11	establish	establish	VERB
ejpam-6813	6	12	structural	structural	ADJ
ejpam-6813	6	13	theorems	theorem	NOUN
ejpam-6813	6	14	for	for	ADP
ejpam-6813	6	15	block	block	NOUN
ejpam-6813	6	16	matrices	matrix	NOUN
ejpam-6813	6	17	.	.	PUNCT
ejpam-6813	7	1	2020	2020	NUM
ejpam-6813	7	2	mathematics	mathematic	NOUN
ejpam-6813	7	3	subject	subject	NOUN
ejpam-6813	7	4	classifications	classification	NOUN
ejpam-6813	7	5	:	:	PUNCT
ejpam-6813	7	6	15a42	15a42	NUM
ejpam-6813	7	7	,	,	PUNCT
ejpam-6813	7	8	15a60	15a60	NUM
ejpam-6813	7	9	,	,	PUNCT
ejpam-6813	7	10	47a63	47a63	NUM
ejpam-6813	7	11	,	,	PUNCT
ejpam-6813	7	12	47b15	47b15	NUM
ejpam-6813	7	13	,	,	PUNCT
ejpam-6813	7	14	47b47	47b47	VERB
ejpam-6813	7	15	key	key	ADJ
ejpam-6813	7	16	words	word	NOUN
ejpam-6813	7	17	and	and	CCONJ
ejpam-6813	7	18	phrases	phrase	NOUN
ejpam-6813	7	19	:	:	PUNCT
ejpam-6813	7	20	numerical	numerical	ADJ
ejpam-6813	7	21	radius	radius	NOUN
ejpam-6813	7	22	,	,	PUNCT
ejpam-6813	7	23	generalized	generalized	ADJ
ejpam-6813	7	24	numerical	numerical	ADJ
ejpam-6813	7	25	radius	radius	NOUN
ejpam-6813	7	26	,	,	PUNCT
ejpam-6813	7	27	block	block	NOUN
ejpam-6813	7	28	matrices	matrix	NOUN
ejpam-6813	7	29	,	,	PUNCT
ejpam-6813	7	30	positive	positive	ADJ
ejpam-6813	7	31	semidefinite	semidefinite	NOUN
ejpam-6813	7	32	matrices	matrix	NOUN
ejpam-6813	7	33	,	,	PUNCT
ejpam-6813	7	34	unitarily	unitarily	ADV
ejpam-6813	7	35	invariant	invariant	ADJ
ejpam-6813	7	36	norm	norm	NOUN
ejpam-6813	7	37	1	1	NUM
ejpam-6813	7	38	.	.	PUNCT
ejpam-6813	7	39	introduction	introduction	NOUN
ejpam-6813	7	40	matrix	matrix	NOUN
ejpam-6813	7	41	analysis	analysis	NOUN
ejpam-6813	7	42	has	have	AUX
ejpam-6813	7	43	witnessed	witness	VERB
ejpam-6813	7	44	remarkable	remarkable	ADJ
ejpam-6813	7	45	progress	progress	NOUN
ejpam-6813	7	46	in	in	ADP
ejpam-6813	7	47	understanding	understand	VERB
ejpam-6813	7	48	the	the	DET
ejpam-6813	7	49	geometric	geometric	ADJ
ejpam-6813	7	50	properties	property	NOUN
ejpam-6813	7	51	of	of	ADP
ejpam-6813	7	52	operators	operator	NOUN
ejpam-6813	7	53	through	through	ADP
ejpam-6813	7	54	the	the	DET
ejpam-6813	7	55	numerical	numerical	ADJ
ejpam-6813	7	56	radius	radius	NOUN
ejpam-6813	7	57	.	.	PUNCT
ejpam-6813	8	1	this	this	DET
ejpam-6813	8	2	functional	functional	ADJ
ejpam-6813	8	3	provides	provide	VERB
ejpam-6813	8	4	valuable	valuable	ADJ
ejpam-6813	8	5	insights	insight	NOUN
ejpam-6813	8	6	into	into	ADP
ejpam-6813	8	7	the	the	DET
ejpam-6813	8	8	field	field	NOUN
ejpam-6813	8	9	of	of	ADP
ejpam-6813	8	10	values	value	NOUN
ejpam-6813	8	11	and	and	CCONJ
ejpam-6813	8	12	offers	offer	VERB
ejpam-6813	8	13	geometric	geometric	ADJ
ejpam-6813	8	14	interpretations	interpretation	NOUN
ejpam-6813	8	15	that	that	PRON
ejpam-6813	8	16	complement	complement	VERB
ejpam-6813	8	17	1corresponding	1corresponding	NUM
ejpam-6813	8	18	author	author	NOUN
ejpam-6813	8	19	.	.	PUNCT
ejpam-6813	9	1	0doi	0doi	PROPN
ejpam-6813	9	2	:	:	PUNCT
ejpam-6813	9	3	https://doi.org/10.29020/nybg.ejpam.v18i4.6813	https://doi.org/10.29020/nybg.ejpam.v18i4.6813	PROPN
ejpam-6813	9	4	email	email	NOUN
ejpam-6813	9	5	addresses	address	NOUN
ejpam-6813	9	6	:	:	PUNCT
ejpam-6813	10	1	rajaa.alnaimi@uop.edu.jo	rajaa.alnaimi@uop.edu.jo	ADJ
ejpam-6813	10	2	,	,	PUNCT
ejpam-6813	10	3	rajaa.alnaimi@eau.ac.ae	rajaa.alnaimi@eau.ac.ae	ADJ
ejpam-6813	10	4	(	(	PUNCT
ejpam-6813	10	5	r.	r.	PROPN
ejpam-6813	10	6	al	al	PROPN
ejpam-6813	10	7	-	-	PUNCT
ejpam-6813	10	8	naimi	naimi	NOUN
ejpam-6813	10	9	)	)	PUNCT
ejpam-6813	10	10	,	,	PUNCT
ejpam-6813	10	11	manal.allabadi@uop.edu.jo	manal.allabadi@uop.edu.jo	X
ejpam-6813	10	12	(	(	PUNCT
ejpam-6813	10	13	m.	m.	NOUN
ejpam-6813	10	14	al	al	PROPN
ejpam-6813	10	15	-	-	PUNCT
ejpam-6813	10	16	labadi	labadi	NOUN
ejpam-6813	10	17	)	)	PUNCT
ejpam-6813	10	18	,	,	PUNCT
ejpam-6813	10	19	waudeh@uop.edu.jo	waudeh@uop.edu.jo	NOUN
ejpam-6813	10	20	(	(	PUNCT
ejpam-6813	10	21	w.	w.	PROPN
ejpam-6813	10	22	audeh	audeh	PROPN
ejpam-6813	10	23	)	)	PUNCT
ejpam-6813	10	24	,	,	PUNCT
ejpam-6813	10	25	jamal.oudetallah@uop.edu.jo	jamal.oudetallah@uop.edu.jo	PROPN
ejpam-6813	10	26	(	(	PUNCT
ejpam-6813	10	27	j.	j.	PROPN
ejpam-6813	10	28	oudetallah	oudetallah	PROPN
ejpam-6813	10	29	)	)	PUNCT
ejpam-6813	10	30	,	,	PUNCT
ejpam-6813	10	31	mutti.rehman@aiu.uz	mutti.rehman@aiu.uz	X
ejpam-6813	10	32	(	(	PUNCT
ejpam-6813	10	33	m.-u	m.-u	PROPN
ejpam-6813	10	34	.	.	PUNCT
ejpam-6813	11	1	rehman	rehman	PROPN
ejpam-6813	11	2	)	)	PUNCT
ejpam-6813	11	3	,	,	PUNCT
ejpam-6813	11	4	dheyaa.alangood@uot.edu.iq	dheyaa.alangood@uot.edu.iq	PROPN
ejpam-6813	11	5	(	(	PUNCT
ejpam-6813	11	6	d.	d.	PROPN
ejpam-6813	11	7	alangood	alangood	PROPN
ejpam-6813	11	8	)	)	PUNCT
ejpam-6813	11	9	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6813	12	1	1	1	NUM
ejpam-6813	12	2	copyright	copyright	NOUN
ejpam-6813	12	3	:	:	PUNCT
ejpam-6813	12	4	©	©	PROPN
ejpam-6813	12	5	2025	2025	NUM
ejpam-6813	12	6	the	the	DET
ejpam-6813	12	7	author(s	author(s	NOUN
ejpam-6813	12	8	)	)	PUNCT
ejpam-6813	12	9	.	.	PUNCT
ejpam-6813	13	1	(	(	PUNCT
ejpam-6813	13	2	cc	cc	NOUN
ejpam-6813	13	3	by	by	ADP
ejpam-6813	13	4	-	-	PUNCT
ejpam-6813	13	5	nc	nc	PROPN
ejpam-6813	13	6	4.0	4.0	NUM
ejpam-6813	13	7	)	)	PUNCT
ejpam-6813	13	8	r.	r.	PROPN
ejpam-6813	13	9	al	al	PROPN
ejpam-6813	13	10	-	-	PROPN
ejpam-6813	13	11	naimi	naimi	PROPN
ejpam-6813	13	12	et	et	PROPN
ejpam-6813	13	13	al	al	PROPN
ejpam-6813	13	14	.	.	PUNCT
ejpam-6813	13	15	/	/	SYM
ejpam-6813	13	16	eur	eur	PROPN
ejpam-6813	13	17	.	.	PUNCT
ejpam-6813	14	1	j.	j.	PROPN
ejpam-6813	14	2	pure	pure	PROPN
ejpam-6813	14	3	appl	appl	PROPN
ejpam-6813	14	4	.	.	PROPN
ejpam-6813	14	5	math	math	PROPN
ejpam-6813	14	6	,	,	PUNCT
ejpam-6813	14	7	18	18	NUM
ejpam-6813	14	8	(	(	PUNCT
ejpam-6813	14	9	4	4	NUM
ejpam-6813	14	10	)	)	PUNCT
ejpam-6813	14	11	(	(	PUNCT
ejpam-6813	14	12	2025	2025	NUM
ejpam-6813	14	13	)	)	PUNCT
ejpam-6813	14	14	,	,	PUNCT
ejpam-6813	14	15	6813	6813	NUM
ejpam-6813	14	16	2	2	NUM
ejpam-6813	14	17	of	of	ADP
ejpam-6813	14	18	15	15	NUM
ejpam-6813	14	19	traditional	traditional	ADJ
ejpam-6813	14	20	spectral	spectral	ADJ
ejpam-6813	14	21	approaches	approach	NOUN
ejpam-6813	14	22	.	.	PUNCT
ejpam-6813	15	1	the	the	DET
ejpam-6813	15	2	numerical	numerical	ADJ
ejpam-6813	15	3	radius	radius	NOUN
ejpam-6813	15	4	,	,	PUNCT
ejpam-6813	15	5	originally	originally	ADV
ejpam-6813	15	6	studied	study	VERB
ejpam-6813	15	7	for	for	ADP
ejpam-6813	15	8	single	single	ADJ
ejpam-6813	15	9	matrices	matrix	NOUN
ejpam-6813	15	10	,	,	PUNCT
ejpam-6813	15	11	has	have	AUX
ejpam-6813	15	12	found	find	VERB
ejpam-6813	15	13	extensive	extensive	ADJ
ejpam-6813	15	14	applications	application	NOUN
ejpam-6813	15	15	in	in	ADP
ejpam-6813	15	16	operator	operator	NOUN
ejpam-6813	15	17	theory	theory	NOUN
ejpam-6813	15	18	,	,	PUNCT
ejpam-6813	15	19	quantum	quantum	NOUN
ejpam-6813	15	20	mechanics	mechanic	NOUN
ejpam-6813	15	21	,	,	PUNCT
ejpam-6813	15	22	and	and	CCONJ
ejpam-6813	15	23	computational	computational	ADJ
ejpam-6813	15	24	mathematics	mathematic	NOUN
ejpam-6813	15	25	.	.	PUNCT
ejpam-6813	16	1	let	let	VERB
ejpam-6813	16	2	mm(mn	mm(mn	NOUN
ejpam-6813	16	3	)	)	PUNCT
ejpam-6813	16	4	denote	denote	VERB
ejpam-6813	16	5	the	the	DET
ejpam-6813	16	6	collection	collection	NOUN
ejpam-6813	16	7	of	of	ADP
ejpam-6813	16	8	all	all	DET
ejpam-6813	16	9	m×m	m×m	ADJ
ejpam-6813	16	10	block	block	NOUN
ejpam-6813	16	11	complex	complex	ADJ
ejpam-6813	16	12	matrices	matrix	NOUN
ejpam-6813	16	13	where	where	SCONJ
ejpam-6813	16	14	each	each	DET
ejpam-6813	16	15	entry	entry	NOUN
ejpam-6813	16	16	belongs	belong	VERB
ejpam-6813	16	17	to	to	ADP
ejpam-6813	16	18	mn(c	mn(c	NOUN
ejpam-6813	16	19	)	)	PUNCT
ejpam-6813	16	20	.	.	PUNCT
ejpam-6813	17	1	when	when	SCONJ
ejpam-6813	17	2	dealing	deal	VERB
ejpam-6813	17	3	with	with	ADP
ejpam-6813	17	4	the	the	DET
ejpam-6813	17	5	special	special	ADJ
ejpam-6813	17	6	case	case	NOUN
ejpam-6813	17	7	mm(c	mm(c	NUM
ejpam-6813	17	8	)	)	PUNCT
ejpam-6813	17	9	,	,	PUNCT
ejpam-6813	17	10	we	we	PRON
ejpam-6813	17	11	have	have	VERB
ejpam-6813	17	12	the	the	DET
ejpam-6813	17	13	standard	standard	ADJ
ejpam-6813	17	14	set	set	NOUN
ejpam-6813	17	15	of	of	ADP
ejpam-6813	17	16	m	m	PROPN
ejpam-6813	17	17	×	×	PROPN
ejpam-6813	17	18	m	m	NOUN
ejpam-6813	17	19	complex	complex	ADJ
ejpam-6813	17	20	matrices	matrix	NOUN
ejpam-6813	17	21	.	.	PUNCT
ejpam-6813	18	1	a	a	DET
ejpam-6813	18	2	matrix	matrix	NOUN
ejpam-6813	18	3	a	a	DET
ejpam-6813	18	4	∈	∈	NOUN
ejpam-6813	18	5	mm(c	mm(c	X
ejpam-6813	18	6	)	)	PUNCT
ejpam-6813	18	7	is	be	AUX
ejpam-6813	18	8	termed	term	VERB
ejpam-6813	18	9	positive	positive	ADJ
ejpam-6813	18	10	semidefinite	semidefinite	NOUN
ejpam-6813	18	11	,	,	PUNCT
ejpam-6813	18	12	written	write	VERB
ejpam-6813	18	13	as	as	ADP
ejpam-6813	18	14	a	a	DET
ejpam-6813	18	15	≥	≥	NOUN
ejpam-6813	18	16	0	0	NUM
ejpam-6813	18	17	,	,	PUNCT
ejpam-6813	18	18	when	when	SCONJ
ejpam-6813	18	19	xtax	xtax	PROPN
ejpam-6813	18	20	≥	≥	NOUN
ejpam-6813	18	21	0	0	NUM
ejpam-6813	18	22	holds	hold	VERB
ejpam-6813	18	23	for	for	ADP
ejpam-6813	18	24	every	every	DET
ejpam-6813	18	25	x	x	PROPN
ejpam-6813	18	26	∈	∈	PROPN
ejpam-6813	18	27	c.	c.	NOUN
ejpam-6813	18	28	given	give	VERB
ejpam-6813	18	29	a	a	DET
ejpam-6813	18	30	∈	∈	NOUN
ejpam-6813	18	31	mm(c	mm(c	NOUN
ejpam-6813	18	32	)	)	PUNCT
ejpam-6813	18	33	,	,	PUNCT
ejpam-6813	18	34	we	we	PRON
ejpam-6813	18	35	use	use	VERB
ejpam-6813	18	36	at	at	ADP
ejpam-6813	18	37	,	,	PUNCT
ejpam-6813	18	38	aτ	aτ	ADV
ejpam-6813	18	39	,	,	PUNCT
ejpam-6813	18	40	and	and	CCONJ
ejpam-6813	18	41	a∗	a∗	PROPN
ejpam-6813	18	42	to	to	PART
ejpam-6813	18	43	represent	represent	VERB
ejpam-6813	18	44	the	the	DET
ejpam-6813	18	45	transpose	transpose	ADJ
ejpam-6813	18	46	,	,	PUNCT
ejpam-6813	18	47	partial	partial	ADJ
ejpam-6813	18	48	transpose	transpose	NOUN
ejpam-6813	18	49	,	,	PUNCT
ejpam-6813	18	50	and	and	CCONJ
ejpam-6813	18	51	conjugate	conjugate	ADJ
ejpam-6813	18	52	transpose	transpose	ADJ
ejpam-6813	18	53	operations	operation	NOUN
ejpam-6813	18	54	,	,	PUNCT
ejpam-6813	18	55	respectively	respectively	ADV
ejpam-6813	18	56	.	.	PUNCT
ejpam-6813	19	1	the	the	DET
ejpam-6813	19	2	absolute	absolute	ADJ
ejpam-6813	19	3	value	value	NOUN
ejpam-6813	19	4	is	be	AUX
ejpam-6813	19	5	expressed	express	VERB
ejpam-6813	19	6	as	as	ADP
ejpam-6813	19	7	|a|	|a|	NOUN
ejpam-6813	19	8	=	=	SYM
ejpam-6813	19	9	(	(	PUNCT
ejpam-6813	19	10	a∗a)1/2	a∗a)1/2	PROPN
ejpam-6813	19	11	.	.	PUNCT
ejpam-6813	20	1	we	we	PRON
ejpam-6813	20	2	denote	denote	VERB
ejpam-6813	20	3	by	by	ADP
ejpam-6813	20	4	σ(a	σ(a	PROPN
ejpam-6813	20	5	)	)	PUNCT
ejpam-6813	20	6	=	=	PRON
ejpam-6813	20	7	{	{	PUNCT
ejpam-6813	20	8	λ1(a	λ1(a	NOUN
ejpam-6813	20	9	)	)	PUNCT
ejpam-6813	20	10	,	,	PUNCT
ejpam-6813	20	11	.	.	PUNCT
ejpam-6813	20	12	.	.	PUNCT
ejpam-6813	20	13	.	.	PUNCT
ejpam-6813	21	1	,	,	PUNCT
ejpam-6813	21	2	λm(a	λm(a	NOUN
ejpam-6813	21	3	)	)	PUNCT
ejpam-6813	21	4	}	}	PUNCT
ejpam-6813	21	5	the	the	DET
ejpam-6813	21	6	complete	complete	ADJ
ejpam-6813	21	7	eigenvalue	eigenvalue	NOUN
ejpam-6813	21	8	set	set	NOUN
ejpam-6813	21	9	of	of	ADP
ejpam-6813	21	10	a	a	DET
ejpam-6813	21	11	arranged	arrange	VERB
ejpam-6813	21	12	such	such	ADJ
ejpam-6813	21	13	that	that	SCONJ
ejpam-6813	21	14	|λ1(a)|	|λ1(a)|	NUM
ejpam-6813	21	15	≥	≥	NOUN
ejpam-6813	21	16	·	·	PUNCT
ejpam-6813	21	17	·	·	PUNCT
ejpam-6813	21	18	·	·	PUNCT
ejpam-6813	21	19	≥	≥	PRON
ejpam-6813	21	20	|λm(a)|	|λm(a)|	NOUN
ejpam-6813	21	21	.	.	PUNCT
ejpam-6813	22	1	the	the	DET
ejpam-6813	22	2	spectral	spectral	ADJ
ejpam-6813	22	3	radius	radius	NOUN
ejpam-6813	22	4	,	,	PUNCT
ejpam-6813	22	5	spectral	spectral	ADJ
ejpam-6813	22	6	norm	norm	NOUN
ejpam-6813	22	7	,	,	PUNCT
ejpam-6813	22	8	and	and	CCONJ
ejpam-6813	22	9	any	any	DET
ejpam-6813	22	10	unitarily	unitarily	ADJ
ejpam-6813	22	11	invariant	invariant	ADJ
ejpam-6813	22	12	norm	norm	NOUN
ejpam-6813	22	13	of	of	ADP
ejpam-6813	22	14	a	a	PRON
ejpam-6813	22	15	are	be	AUX
ejpam-6813	22	16	represented	represent	VERB
ejpam-6813	22	17	by	by	ADP
ejpam-6813	22	18	r(a	r(a	PROPN
ejpam-6813	22	19	)	)	PUNCT
ejpam-6813	22	20	,	,	PUNCT
ejpam-6813	22	21	‖a‖	‖a‖	PROPN
ejpam-6813	22	22	,	,	PUNCT
ejpam-6813	22	23	and	and	CCONJ
ejpam-6813	22	24	n(a	n(a	NOUN
ejpam-6813	22	25	)	)	PUNCT
ejpam-6813	22	26	respectively	respectively	ADV
ejpam-6813	22	27	.	.	PUNCT
ejpam-6813	23	1	for	for	ADP
ejpam-6813	23	2	any	any	DET
ejpam-6813	23	3	matrix	matrix	NOUN
ejpam-6813	23	4	a	a	DET
ejpam-6813	23	5	∈	∈	NOUN
ejpam-6813	23	6	mm(c	mm(c	NOUN
ejpam-6813	23	7	)	)	PUNCT
ejpam-6813	23	8	,	,	PUNCT
ejpam-6813	23	9	the	the	DET
ejpam-6813	23	10	spectral	spectral	ADJ
ejpam-6813	23	11	radius	radius	NOUN
ejpam-6813	23	12	is	be	AUX
ejpam-6813	23	13	given	give	VERB
ejpam-6813	23	14	by	by	ADP
ejpam-6813	23	15	r(a	r(a	ADJ
ejpam-6813	23	16	)	)	PUNCT
ejpam-6813	23	17	=	=	VERB
ejpam-6813	23	18	|λ1|	|λ1|	NOUN
ejpam-6813	23	19	,	,	PUNCT
ejpam-6813	23	20	while	while	SCONJ
ejpam-6813	23	21	the	the	DET
ejpam-6813	23	22	spectral	spectral	ADJ
ejpam-6813	23	23	norm	norm	NOUN
ejpam-6813	23	24	is	be	AUX
ejpam-6813	23	25	defined	define	VERB
ejpam-6813	23	26	through	through	ADP
ejpam-6813	23	27	‖a‖	‖a‖	PROPN
ejpam-6813	23	28	=	=	SYM
ejpam-6813	23	29	max	max	PROPN
ejpam-6813	23	30	‖x‖=1	‖x‖=1	PROPN
ejpam-6813	23	31	‖ax‖.	‖ax‖.	VERB
ejpam-6813	23	32	the	the	DET
ejpam-6813	23	33	numerical	numerical	PROPN
ejpam-6813	23	34	radius	radius	NOUN
ejpam-6813	23	35	w(a	w(a	VERB
ejpam-6813	23	36	)	)	PUNCT
ejpam-6813	23	37	and	and	CCONJ
ejpam-6813	23	38	generalized	generalized	ADJ
ejpam-6813	23	39	numerical	numerical	ADJ
ejpam-6813	23	40	radius	radius	PROPN
ejpam-6813	23	41	wn	wn	PROPN
ejpam-6813	23	42	(	(	PUNCT
ejpam-6813	23	43	a	a	NOUN
ejpam-6813	23	44	)	)	PUNCT
ejpam-6813	23	45	of	of	ADP
ejpam-6813	23	46	a	a	DET
ejpam-6813	23	47	matrix	matrix	NOUN
ejpam-6813	23	48	a	a	DET
ejpam-6813	23	49	play	play	NOUN
ejpam-6813	23	50	central	central	ADJ
ejpam-6813	23	51	roles	role	NOUN
ejpam-6813	23	52	in	in	ADP
ejpam-6813	23	53	our	our	PRON
ejpam-6813	23	54	analysis	analysis	NOUN
ejpam-6813	23	55	.	.	PUNCT
ejpam-6813	24	1	specifically	specifically	ADV
ejpam-6813	24	2	,	,	PUNCT
ejpam-6813	24	3	for	for	ADP
ejpam-6813	24	4	a	a	DET
ejpam-6813	24	5	∈	∈	NOUN
ejpam-6813	24	6	mm(c	mm(c	NOUN
ejpam-6813	24	7	)	)	PUNCT
ejpam-6813	24	8	,	,	PUNCT
ejpam-6813	24	9	the	the	DET
ejpam-6813	24	10	numerical	numerical	ADJ
ejpam-6813	24	11	radius	radius	NOUN
ejpam-6813	24	12	is	be	AUX
ejpam-6813	24	13	characterized	characterize	VERB
ejpam-6813	24	14	by	by	ADP
ejpam-6813	24	15	w(a	w(a	NOUN
ejpam-6813	24	16	)	)	PUNCT
ejpam-6813	24	17	=	=	SYM
ejpam-6813	24	18	max	max	PROPN
ejpam-6813	24	19	‖x‖=1	‖x‖=1	PROPN
ejpam-6813	24	20	|〈ax	|〈ax	NOUN
ejpam-6813	24	21	,	,	PUNCT
ejpam-6813	24	22	x〉|	x〉|	PROPN
ejpam-6813	24	23	,	,	PUNCT
ejpam-6813	24	24	while	while	SCONJ
ejpam-6813	24	25	the	the	DET
ejpam-6813	24	26	generalized	generalized	ADJ
ejpam-6813	24	27	numerical	numerical	ADJ
ejpam-6813	24	28	radius	radius	NOUN
ejpam-6813	24	29	takes	take	VERB
ejpam-6813	24	30	the	the	DET
ejpam-6813	24	31	form	form	NOUN
ejpam-6813	24	32	wn	wn	X
ejpam-6813	24	33	(	(	PUNCT
ejpam-6813	24	34	a	a	NOUN
ejpam-6813	24	35	)	)	PUNCT
ejpam-6813	24	36	=	=	SYM
ejpam-6813	24	37	max	max	PROPN
ejpam-6813	24	38	θ∈r	θ∈r	ADV
ejpam-6813	24	39	n(re(eiθa	n(re(eiθa	PROPN
ejpam-6813	24	40	)	)	PUNCT
ejpam-6813	24	41	)	)	PUNCT
ejpam-6813	24	42	.	.	PUNCT
ejpam-6813	25	1	(	(	PUNCT
ejpam-6813	25	2	1	1	X
ejpam-6813	25	3	)	)	PUNCT
ejpam-6813	25	4	when	when	SCONJ
ejpam-6813	25	5	we	we	PRON
ejpam-6813	25	6	set	set	VERB
ejpam-6813	25	7	n	n	CCONJ
ejpam-6813	25	8	(	(	PUNCT
ejpam-6813	25	9	·	·	PUNCT
ejpam-6813	25	10	)	)	PUNCT
ejpam-6813	25	11	=	=	PUNCT
ejpam-6813	25	12	‖	‖	PROPN
ejpam-6813	25	13	·	·	PUNCT
ejpam-6813	25	14	‖	‖	VERB
ejpam-6813	25	15	in	in	ADP
ejpam-6813	25	16	equation	equation	NOUN
ejpam-6813	25	17	(	(	PUNCT
ejpam-6813	25	18	1	1	NUM
ejpam-6813	25	19	)	)	PUNCT
ejpam-6813	25	20	,	,	PUNCT
ejpam-6813	25	21	we	we	PRON
ejpam-6813	25	22	recover	recover	VERB
ejpam-6813	25	23	the	the	DET
ejpam-6813	25	24	standard	standard	ADJ
ejpam-6813	25	25	numerical	numerical	ADJ
ejpam-6813	25	26	radius	radius	NOUN
ejpam-6813	25	27	:	:	PUNCT
ejpam-6813	25	28	w(a	w(a	X
ejpam-6813	25	29	)	)	PUNCT
ejpam-6813	26	1	=	=	SYM
ejpam-6813	26	2	max	max	X
ejpam-6813	26	3	θ∈r	θ∈r	ADV
ejpam-6813	26	4	‖re(eiθa)‖.	‖re(eiθa)‖.	X
ejpam-6813	26	5	(	(	PUNCT
ejpam-6813	26	6	2	2	NUM
ejpam-6813	26	7	)	)	PUNCT
ejpam-6813	26	8	a	a	DET
ejpam-6813	26	9	fundamental	fundamental	ADJ
ejpam-6813	26	10	relationship	relationship	NOUN
ejpam-6813	26	11	exists	exist	VERB
ejpam-6813	26	12	for	for	ADP
ejpam-6813	26	13	normal	normal	ADJ
ejpam-6813	26	14	matrices	matrix	NOUN
ejpam-6813	26	15	a	a	DET
ejpam-6813	26	16	∈	∈	NOUN
ejpam-6813	26	17	mn(c	mn(c	X
ejpam-6813	26	18	):	):	PUNCT
ejpam-6813	26	19	w(a	w(a	X
ejpam-6813	26	20	)	)	PUNCT
ejpam-6813	26	21	=	=	SYM
ejpam-6813	27	1	‖a‖	‖a‖	PROPN
ejpam-6813	27	2	=	=	PUNCT
ejpam-6813	27	3	r(a	r(a	PROPN
ejpam-6813	27	4	)	)	PUNCT
ejpam-6813	27	5	.	.	PUNCT
ejpam-6813	28	1	(	(	PUNCT
ejpam-6813	28	2	3	3	X
ejpam-6813	28	3	)	)	PUNCT
ejpam-6813	28	4	for	for	ADP
ejpam-6813	28	5	comprehensive	comprehensive	ADJ
ejpam-6813	28	6	treatments	treatment	NOUN
ejpam-6813	28	7	of	of	ADP
ejpam-6813	28	8	numerical	numerical	ADJ
ejpam-6813	28	9	radius	radius	NOUN
ejpam-6813	28	10	properties	property	NOUN
ejpam-6813	28	11	and	and	CCONJ
ejpam-6813	28	12	generalized	generalized	ADJ
ejpam-6813	28	13	versions	version	NOUN
ejpam-6813	28	14	,	,	PUNCT
ejpam-6813	28	15	readers	reader	NOUN
ejpam-6813	28	16	are	be	AUX
ejpam-6813	28	17	directed	direct	VERB
ejpam-6813	28	18	to	to	ADP
ejpam-6813	28	19	[	[	X
ejpam-6813	28	20	1	1	NUM
ejpam-6813	28	21	]	]	PUNCT
ejpam-6813	28	22	,	,	PUNCT
ejpam-6813	29	1	[	[	X
ejpam-6813	29	2	2	2	NUM
ejpam-6813	29	3	]	]	PUNCT
ejpam-6813	29	4	,	,	PUNCT
ejpam-6813	29	5	and	and	CCONJ
ejpam-6813	29	6	[	[	X
ejpam-6813	29	7	3	3	NUM
ejpam-6813	29	8	]	]	PUNCT
ejpam-6813	29	9	.	.	PUNCT
ejpam-6813	30	1	the	the	DET
ejpam-6813	30	2	connection	connection	NOUN
ejpam-6813	30	3	to	to	ADP
ejpam-6813	30	4	singular	singular	ADJ
ejpam-6813	30	5	values	value	NOUN
ejpam-6813	30	6	is	be	AUX
ejpam-6813	30	7	established	establish	VERB
ejpam-6813	30	8	through	through	ADP
ejpam-6813	30	9	‖a‖	‖a‖	PROPN
ejpam-6813	30	10	=	=	PUNCT
ejpam-6813	30	11	s1(a	s1(a	PROPN
ejpam-6813	30	12	)	)	PUNCT
ejpam-6813	30	13	,	,	PUNCT
ejpam-6813	30	14	where	where	SCONJ
ejpam-6813	30	15	s1(a	s1(a	VERB
ejpam-6813	30	16	)	)	PUNCT
ejpam-6813	30	17	represents	represent	VERB
ejpam-6813	30	18	the	the	DET
ejpam-6813	30	19	largest	large	ADJ
ejpam-6813	30	20	singular	singular	ADJ
ejpam-6813	30	21	value	value	NOUN
ejpam-6813	30	22	of	of	ADP
ejpam-6813	30	23	matrix	matrix	NOUN
ejpam-6813	30	24	a	a	DET
ejpam-6813	30	25	∈	∈	NOUN
ejpam-6813	30	26	mm(c	mm(c	NOUN
ejpam-6813	30	27	)	)	PUNCT
ejpam-6813	30	28	.	.	PUNCT
ejpam-6813	31	1	these	these	DET
ejpam-6813	31	2	singular	singular	ADJ
ejpam-6813	31	3	values	value	NOUN
ejpam-6813	31	4	follow	follow	VERB
ejpam-6813	31	5	the	the	DET
ejpam-6813	31	6	ordering	ordering	NOUN
ejpam-6813	31	7	s1(a	s1(a	ADP
ejpam-6813	31	8	)	)	PUNCT
ejpam-6813	31	9	≥	≥	NOUN
ejpam-6813	31	10	s2(a	s2(a	NOUN
ejpam-6813	31	11	)	)	PUNCT
ejpam-6813	31	12	≥	≥	NOUN
ejpam-6813	31	13	·	·	PUNCT
ejpam-6813	31	14	·	·	PUNCT
ejpam-6813	31	15	·	·	PUNCT
ejpam-6813	31	16	≥	≥	NUM
ejpam-6813	31	17	sm(a	sm(a	NOUN
ejpam-6813	31	18	)	)	PUNCT
ejpam-6813	31	19	and	and	CCONJ
ejpam-6813	31	20	are	be	AUX
ejpam-6813	31	21	defined	define	VERB
ejpam-6813	31	22	via	via	ADP
ejpam-6813	31	23	sj(a	sj(a	NOUN
ejpam-6813	31	24	)	)	PUNCT
ejpam-6813	31	25	=	=	SYM
ejpam-6813	31	26	λj(|a|	λj(|a|	ADJ
ejpam-6813	31	27	)	)	PUNCT
ejpam-6813	31	28	for	for	ADP
ejpam-6813	31	29	j	j	PROPN
ejpam-6813	31	30	=	=	SYM
ejpam-6813	31	31	1	1	NUM
ejpam-6813	31	32	,	,	PUNCT
ejpam-6813	31	33	2	2	NUM
ejpam-6813	31	34	,	,	PUNCT
ejpam-6813	31	35	.	.	PUNCT
ejpam-6813	31	36	.	.	PUNCT
ejpam-6813	32	1	.	.	PUNCT
ejpam-6813	33	1	,	,	PUNCT
ejpam-6813	33	2	m.	m.	NOUN
ejpam-6813	33	3	comprehensive	comprehensive	ADJ
ejpam-6813	33	4	discussions	discussion	NOUN
ejpam-6813	33	5	of	of	ADP
ejpam-6813	33	6	singular	singular	ADJ
ejpam-6813	33	7	value	value	NOUN
ejpam-6813	33	8	theory	theory	NOUN
ejpam-6813	33	9	can	can	AUX
ejpam-6813	33	10	be	be	AUX
ejpam-6813	33	11	found	find	VERB
ejpam-6813	33	12	in	in	ADP
ejpam-6813	33	13	[	[	X
ejpam-6813	33	14	4–8	4–8	NOUN
ejpam-6813	33	15	]	]	X
ejpam-6813	33	16	.	.	PUNCT
ejpam-6813	34	1	when	when	SCONJ
ejpam-6813	34	2	working	work	VERB
ejpam-6813	34	3	with	with	ADP
ejpam-6813	34	4	matrices	matrix	NOUN
ejpam-6813	34	5	a	a	PRON
ejpam-6813	34	6	=	=	NOUN
ejpam-6813	34	7	[	[	X
ejpam-6813	34	8	ai	ai	NOUN
ejpam-6813	34	9	,	,	PUNCT
ejpam-6813	34	10	j	j	PROPN
ejpam-6813	34	11	]	]	PUNCT
ejpam-6813	34	12	∈	∈	PROPN
ejpam-6813	34	13	mm(mn	mm(mn	PROPN
ejpam-6813	34	14	)	)	PUNCT
ejpam-6813	34	15	and	and	CCONJ
ejpam-6813	34	16	b	b	X
ejpam-6813	34	17	=	=	SYM
ejpam-6813	35	1	[	[	X
ejpam-6813	35	2	bi	bi	NOUN
ejpam-6813	35	3	,	,	PUNCT
ejpam-6813	35	4	j	j	PROPN
ejpam-6813	35	5	]	]	PUNCT
ejpam-6813	35	6	∈	∈	PROPN
ejpam-6813	35	7	mm(mn	mm(mn	PROPN
ejpam-6813	35	8	)	)	PUNCT
ejpam-6813	35	9	of	of	ADP
ejpam-6813	35	10	identical	identical	ADJ
ejpam-6813	35	11	dimensions	dimension	NOUN
ejpam-6813	35	12	,	,	PUNCT
ejpam-6813	35	13	the	the	DET
ejpam-6813	35	14	hadamard	hadamard	ADJ
ejpam-6813	35	15	product	product	NOUN
ejpam-6813	35	16	,	,	PUNCT
ejpam-6813	35	17	as	as	SCONJ
ejpam-6813	35	18	introduced	introduce	VERB
ejpam-6813	35	19	in	in	ADP
ejpam-6813	35	20	[	[	X
ejpam-6813	35	21	9	9	NUM
ejpam-6813	35	22	]	]	PUNCT
ejpam-6813	35	23	,	,	PUNCT
ejpam-6813	35	24	is	be	AUX
ejpam-6813	35	25	given	give	VERB
ejpam-6813	35	26	by	by	ADP
ejpam-6813	35	27	a	a	DET
ejpam-6813	35	28	◦	◦	NOUN
ejpam-6813	35	29	b	b	NOUN
ejpam-6813	36	1	=	=	PUNCT
ejpam-6813	37	1	[	[	X
ejpam-6813	37	2	ai	ai	ADJ
ejpam-6813	37	3	,	,	PUNCT
ejpam-6813	37	4	jbi	jbi	PROPN
ejpam-6813	37	5	,	,	PUNCT
ejpam-6813	37	6	j	j	PROPN
ejpam-6813	37	7	]	]	PUNCT
ejpam-6813	37	8	.	.	PUNCT
ejpam-6813	38	1	the	the	DET
ejpam-6813	38	2	concept	concept	NOUN
ejpam-6813	38	3	of	of	ADP
ejpam-6813	38	4	hadamard	hadamard	ADJ
ejpam-6813	38	5	powers	power	NOUN
ejpam-6813	38	6	,	,	PUNCT
ejpam-6813	38	7	detailed	detail	VERB
ejpam-6813	38	8	in	in	ADP
ejpam-6813	38	9	[	[	X
ejpam-6813	38	10	10	10	NUM
ejpam-6813	38	11	]	]	PUNCT
ejpam-6813	38	12	,	,	PUNCT
ejpam-6813	38	13	is	be	AUX
ejpam-6813	38	14	expressed	express	VERB
ejpam-6813	38	15	as	as	ADP
ejpam-6813	38	16	:	:	PUNCT
ejpam-6813	38	17	a	a	DET
ejpam-6813	38	18	◦	◦	NOUN
ejpam-6813	38	19	n	n	NOUN
ejpam-6813	38	20	=	=	PUNCT
ejpam-6813	39	1	[	[	X
ejpam-6813	39	2	an	an	X
ejpam-6813	39	3	i	i	PROPN
ejpam-6813	39	4	,	,	PUNCT
ejpam-6813	39	5	j	j	PROPN
ejpam-6813	39	6	]	]	X
ejpam-6813	39	7	,	,	PUNCT
ejpam-6813	39	8	(	(	PUNCT
ejpam-6813	39	9	4	4	X
ejpam-6813	39	10	)	)	PUNCT
ejpam-6813	39	11	r.	r.	PROPN
ejpam-6813	39	12	al	al	PROPN
ejpam-6813	39	13	-	-	PROPN
ejpam-6813	39	14	naimi	naimi	PROPN
ejpam-6813	39	15	et	et	PROPN
ejpam-6813	39	16	al	al	PROPN
ejpam-6813	39	17	.	.	PUNCT
ejpam-6813	39	18	/	/	SYM
ejpam-6813	39	19	eur	eur	PROPN
ejpam-6813	39	20	.	.	PUNCT
ejpam-6813	40	1	j.	j.	PROPN
ejpam-6813	40	2	pure	pure	PROPN
ejpam-6813	40	3	appl	appl	PROPN
ejpam-6813	40	4	.	.	PROPN
ejpam-6813	40	5	math	math	PROPN
ejpam-6813	40	6	,	,	PUNCT
ejpam-6813	40	7	18	18	NUM
ejpam-6813	40	8	(	(	PUNCT
ejpam-6813	40	9	4	4	NUM
ejpam-6813	40	10	)	)	PUNCT
ejpam-6813	40	11	(	(	PUNCT
ejpam-6813	40	12	2025	2025	NUM
ejpam-6813	40	13	)	)	PUNCT
ejpam-6813	40	14	,	,	PUNCT
ejpam-6813	40	15	6813	6813	NUM
ejpam-6813	40	16	3	3	NUM
ejpam-6813	40	17	of	of	ADP
ejpam-6813	40	18	15	15	NUM
ejpam-6813	40	19	with	with	ADP
ejpam-6813	40	20	the	the	DET
ejpam-6813	40	21	corresponding	correspond	VERB
ejpam-6813	40	22	hadamard	hadamard	ADJ
ejpam-6813	40	23	inverse	inverse	NOUN
ejpam-6813	40	24	defined	define	VERB
ejpam-6813	40	25	by	by	ADP
ejpam-6813	40	26	:	:	PUNCT
ejpam-6813	40	27	a	a	DET
ejpam-6813	40	28	◦	◦	NOUN
ejpam-6813	40	29	(−1	(−1	NOUN
ejpam-6813	40	30	)	)	PUNCT
ejpam-6813	41	1	=	=	PUNCT
ejpam-6813	42	1	[	[	X
ejpam-6813	42	2	a−1	a−1	PROPN
ejpam-6813	42	3	i	i	PROPN
ejpam-6813	42	4	,	,	PUNCT
ejpam-6813	42	5	j	j	PROPN
ejpam-6813	42	6	]	]	X
ejpam-6813	42	7	.	.	PUNCT
ejpam-6813	43	1	(	(	PUNCT
ejpam-6813	43	2	5	5	NUM
ejpam-6813	43	3	)	)	PUNCT
ejpam-6813	43	4	for	for	ADP
ejpam-6813	43	5	matrices	matrix	NOUN
ejpam-6813	43	6	of	of	ADP
ejpam-6813	43	7	the	the	DET
ejpam-6813	43	8	form	form	NOUN
ejpam-6813	43	9	a	a	PRON
ejpam-6813	43	10	=	=	X
ejpam-6813	44	1	[	[	X
ejpam-6813	44	2	aij	aij	X
ejpam-6813	44	3	]	]	PUNCT
ejpam-6813	44	4	m	m	VERB
ejpam-6813	44	5	i	i	PRON
ejpam-6813	44	6	,	,	PUNCT
ejpam-6813	44	7	j=1	j=1	PROPN
ejpam-6813	44	8	∈	∈	PROPN
ejpam-6813	44	9	mm(c	mm(c	X
ejpam-6813	44	10	)	)	PUNCT
ejpam-6813	44	11	,	,	PUNCT
ejpam-6813	44	12	reference	reference	NOUN
ejpam-6813	44	13	[	[	X
ejpam-6813	44	14	11	11	NUM
ejpam-6813	44	15	]	]	PUNCT
ejpam-6813	44	16	provides	provide	VERB
ejpam-6813	44	17	:	:	PUNCT
ejpam-6813	44	18	a	a	DET
ejpam-6813	44	19	◦	◦	NOUN
ejpam-6813	44	20	(1	(1	NOUN
ejpam-6813	44	21	/	/	SYM
ejpam-6813	44	22	n	n	CCONJ
ejpam-6813	44	23	)	)	PUNCT
ejpam-6813	44	24	=	=	PUNCT
ejpam-6813	45	1	[	[	X
ejpam-6813	45	2	a	a	DET
ejpam-6813	45	3	1	1	NUM
ejpam-6813	45	4	/	/	SYM
ejpam-6813	45	5	n	n	NOUN
ejpam-6813	45	6	i	i	PROPN
ejpam-6813	45	7	,	,	PUNCT
ejpam-6813	45	8	j	j	PROPN
ejpam-6813	45	9	]	]	X
ejpam-6813	45	10	.	.	PUNCT
ejpam-6813	46	1	(	(	PUNCT
ejpam-6813	46	2	6	6	NUM
ejpam-6813	46	3	)	)	PUNCT
ejpam-6813	46	4	research	research	NOUN
ejpam-6813	46	5	findings	finding	NOUN
ejpam-6813	46	6	presented	present	VERB
ejpam-6813	46	7	in	in	ADP
ejpam-6813	46	8	[	[	X
ejpam-6813	46	9	12	12	NUM
ejpam-6813	46	10	]	]	PUNCT
ejpam-6813	46	11	establish	establish	VERB
ejpam-6813	46	12	that	that	SCONJ
ejpam-6813	46	13	for	for	ADP
ejpam-6813	46	14	a	a	DET
ejpam-6813	46	15	∈	∈	PROPN
ejpam-6813	46	16	mm(mn	mm(mn	NOUN
ejpam-6813	46	17	):	):	PUNCT
ejpam-6813	46	18	r(a	r(a	NUM
ejpam-6813	46	19	)	)	PUNCT
ejpam-6813	46	20	≤	≤	NOUN
ejpam-6813	46	21	r([‖ai	r([‖ai	NOUN
ejpam-6813	46	22	,	,	PUNCT
ejpam-6813	46	23	j‖	j‖	PROPN
ejpam-6813	46	24	]	]	PUNCT
ejpam-6813	46	25	)	)	PUNCT
ejpam-6813	46	26	,	,	PUNCT
ejpam-6813	46	27	‖a‖	‖a‖	PROPN
ejpam-6813	46	28	≤	≤	NOUN
ejpam-6813	46	29	‖[‖ai	‖[‖ai	ADP
ejpam-6813	46	30	,	,	PUNCT
ejpam-6813	46	31	j‖]‖	j‖]‖	PROPN
ejpam-6813	46	32	and	and	CCONJ
ejpam-6813	46	33	w(a	w(a	ADJ
ejpam-6813	46	34	)	)	PUNCT
ejpam-6813	46	35	≤	≤	NOUN
ejpam-6813	46	36	r([‖ai	r([‖ai	NOUN
ejpam-6813	46	37	,	,	PUNCT
ejpam-6813	46	38	j‖	j‖	PROPN
ejpam-6813	46	39	]	]	PUNCT
ejpam-6813	46	40	)	)	PUNCT
ejpam-6813	46	41	.	.	PUNCT
ejpam-6813	47	1	(	(	PUNCT
ejpam-6813	47	2	7	7	X
ejpam-6813	47	3	)	)	PUNCT
ejpam-6813	47	4	consider	consider	VERB
ejpam-6813	47	5	a	a	DET
ejpam-6813	47	6	matrix	matrix	NOUN
ejpam-6813	47	7	a	a	PRON
ejpam-6813	47	8	=	=	X
ejpam-6813	48	1	[	[	X
ejpam-6813	48	2	[	[	X
ejpam-6813	48	3	ai	ai	NOUN
ejpam-6813	48	4	,	,	PUNCT
ejpam-6813	48	5	jl	jl	NOUN
ejpam-6813	48	6	,	,	PUNCT
ejpam-6813	48	7	k	k	X
ejpam-6813	48	8	]	]	X
ejpam-6813	48	9	n	n	PRON
ejpam-6813	48	10	l	l	NOUN
ejpam-6813	48	11	,	,	PUNCT
ejpam-6813	48	12	k=1	k=1	X
ejpam-6813	48	13	]	]	X
ejpam-6813	48	14	m	m	VERB
ejpam-6813	48	15	i	i	PRON
ejpam-6813	48	16	,	,	PUNCT
ejpam-6813	48	17	j=1	j=1	PROPN
ejpam-6813	48	18	∈	∈	PROPN
ejpam-6813	48	19	mm(mn	mm(mn	PROPN
ejpam-6813	48	20	)	)	PUNCT
ejpam-6813	48	21	.	.	PUNCT
ejpam-6813	49	1	we	we	PRON
ejpam-6813	49	2	can	can	AUX
ejpam-6813	49	3	construct	construct	VERB
ejpam-6813	49	4	the	the	DET
ejpam-6813	49	5	associated	associate	VERB
ejpam-6813	49	6	matrix	matrix	NOUN
ejpam-6813	49	7	ã	ã	PROPN
ejpam-6813	49	8	∈	∈	PROPN
ejpam-6813	49	9	mn(mm	mn(mm	NOUN
ejpam-6813	49	10	)	)	PUNCT
ejpam-6813	49	11	through	through	ADP
ejpam-6813	49	12	ã	ã	PROPN
ejpam-6813	49	13	=	=	SYM
ejpam-6813	50	1	[	[	X
ejpam-6813	50	2	gl	gl	INTJ
ejpam-6813	50	3	,	,	PUNCT
ejpam-6813	50	4	k	k	X
ejpam-6813	50	5	]	]	X
ejpam-6813	50	6	n	n	PRON
ejpam-6813	50	7	l	l	NOUN
ejpam-6813	50	8	,	,	PUNCT
ejpam-6813	50	9	k=1	k=1	PUNCT
ejpam-6813	50	10	=	=	PUNCT
ejpam-6813	51	1	[	[	X
ejpam-6813	51	2	[	[	X
ejpam-6813	51	3	ai	ai	NOUN
ejpam-6813	51	4	,	,	PUNCT
ejpam-6813	51	5	jl	jl	NOUN
ejpam-6813	51	6	,	,	PUNCT
ejpam-6813	51	7	k	k	X
ejpam-6813	51	8	]	]	X
ejpam-6813	51	9	m	m	VERB
ejpam-6813	51	10	i	i	PRON
ejpam-6813	51	11	,	,	PUNCT
ejpam-6813	51	12	j=1	j=1	X
ejpam-6813	51	13	]	]	X
ejpam-6813	51	14	n	n	PRON
ejpam-6813	51	15	l	l	NOUN
ejpam-6813	51	16	,	,	PUNCT
ejpam-6813	51	17	k=1	k=1	X
ejpam-6813	51	18	,	,	PUNCT
ejpam-6813	51	19	where	where	SCONJ
ejpam-6813	51	20	each	each	DET
ejpam-6813	51	21	gl	gl	NOUN
ejpam-6813	51	22	,	,	PUNCT
ejpam-6813	51	23	k	k	PROPN
ejpam-6813	51	24	=	=	PUNCT
ejpam-6813	52	1	[	[	X
ejpam-6813	52	2	ai	ai	NOUN
ejpam-6813	52	3	,	,	PUNCT
ejpam-6813	52	4	jl	jl	NOUN
ejpam-6813	52	5	,	,	PUNCT
ejpam-6813	52	6	k	k	X
ejpam-6813	52	7	]	]	X
ejpam-6813	52	8	m	m	VERB
ejpam-6813	52	9	i	i	PRON
ejpam-6813	52	10	,	,	PUNCT
ejpam-6813	52	11	j=1	j=1	PROPN
ejpam-6813	52	12	and	and	CCONJ
ejpam-6813	52	13	the	the	DET
ejpam-6813	52	14	relationship	relationship	NOUN
ejpam-6813	52	15	˜̃a	˜̃a	NOUN
ejpam-6813	52	16	=	=	PUNCT
ejpam-6813	52	17	a	a	DET
ejpam-6813	52	18	holds	hold	NOUN
ejpam-6813	52	19	.	.	PUNCT
ejpam-6813	53	1	building	build	VERB
ejpam-6813	53	2	upon	upon	SCONJ
ejpam-6813	53	3	our	our	PRON
ejpam-6813	53	4	previous	previous	ADJ
ejpam-6813	53	5	investigations	investigation	NOUN
ejpam-6813	53	6	into	into	ADP
ejpam-6813	53	7	spectral	spectral	ADJ
ejpam-6813	53	8	properties	property	NOUN
ejpam-6813	53	9	of	of	ADP
ejpam-6813	53	10	block	block	NOUN
ejpam-6813	53	11	matrices	matrix	NOUN
ejpam-6813	53	12	through	through	ADP
ejpam-6813	53	13	partial	partial	ADJ
ejpam-6813	53	14	eigenvalues	eigenvalue	NOUN
ejpam-6813	53	15	[	[	X
ejpam-6813	53	16	13	13	NUM
ejpam-6813	53	17	]	]	PUNCT
ejpam-6813	53	18	,	,	PUNCT
ejpam-6813	53	19	we	we	PRON
ejpam-6813	53	20	now	now	ADV
ejpam-6813	53	21	explore	explore	VERB
ejpam-6813	53	22	complementary	complementary	ADJ
ejpam-6813	53	23	geometric	geometric	ADJ
ejpam-6813	53	24	aspects	aspect	NOUN
ejpam-6813	53	25	via	via	ADP
ejpam-6813	53	26	numerical	numerical	ADJ
ejpam-6813	53	27	radius	radius	PROPN
ejpam-6813	53	28	analysis	analysis	NOUN
ejpam-6813	53	29	.	.	PUNCT
ejpam-6813	54	1	while	while	SCONJ
ejpam-6813	54	2	eigenvalue	eigenvalue	NOUN
ejpam-6813	54	3	-	-	PUNCT
ejpam-6813	54	4	based	base	VERB
ejpam-6813	54	5	methods	method	NOUN
ejpam-6813	54	6	focus	focus	VERB
ejpam-6813	54	7	on	on	ADP
ejpam-6813	54	8	spectral	spectral	ADJ
ejpam-6813	54	9	decomposition	decomposition	NOUN
ejpam-6813	54	10	and	and	CCONJ
ejpam-6813	54	11	diagonalization	diagonalization	NOUN
ejpam-6813	54	12	properties	property	NOUN
ejpam-6813	54	13	,	,	PUNCT
ejpam-6813	54	14	the	the	DET
ejpam-6813	54	15	numerical	numerical	PROPN
ejpam-6813	54	16	radius	radius	PROPN
ejpam-6813	54	17	approach	approach	NOUN
ejpam-6813	54	18	emphasizes	emphasize	VERB
ejpam-6813	54	19	field	field	NOUN
ejpam-6813	54	20	of	of	ADP
ejpam-6813	54	21	values	value	NOUN
ejpam-6813	54	22	and	and	CCONJ
ejpam-6813	54	23	geometric	geometric	ADJ
ejpam-6813	54	24	characterizations	characterization	NOUN
ejpam-6813	54	25	of	of	ADP
ejpam-6813	54	26	matrix	matrix	NOUN
ejpam-6813	54	27	behavior	behavior	NOUN
ejpam-6813	54	28	.	.	PUNCT
ejpam-6813	55	1	in	in	ADP
ejpam-6813	55	2	the	the	DET
ejpam-6813	55	3	present	present	ADJ
ejpam-6813	55	4	work	work	NOUN
ejpam-6813	55	5	,	,	PUNCT
ejpam-6813	55	6	we	we	PRON
ejpam-6813	55	7	introduce	introduce	VERB
ejpam-6813	55	8	novel	novel	ADJ
ejpam-6813	55	9	constructions	construction	NOUN
ejpam-6813	55	10	for	for	ADP
ejpam-6813	55	11	partial	partial	ADJ
ejpam-6813	55	12	matrices	matrix	NOUN
ejpam-6813	55	13	involving	involve	VERB
ejpam-6813	55	14	generalized	generalized	ADJ
ejpam-6813	55	15	numerical	numerical	ADJ
ejpam-6813	55	16	radius	radius	NOUN
ejpam-6813	55	17	and	and	CCONJ
ejpam-6813	55	18	standard	standard	ADJ
ejpam-6813	55	19	numerical	numerical	ADJ
ejpam-6813	55	20	radius	radius	NOUN
ejpam-6813	55	21	:	:	PUNCT
ejpam-6813	55	22	w	w	X
ejpam-6813	55	23	(	(	PUNCT
ejpam-6813	55	24	1	1	NUM
ejpam-6813	55	25	)	)	PUNCT
ejpam-6813	55	26	n	n	CCONJ
ejpam-6813	55	27	(	(	PUNCT
ejpam-6813	55	28	a	a	X
ejpam-6813	55	29	)	)	PUNCT
ejpam-6813	55	30	=	=	PUNCT
ejpam-6813	56	1	[	[	X
ejpam-6813	56	2	wn	wn	X
ejpam-6813	56	3	(	(	PUNCT
ejpam-6813	56	4	gl	gl	PROPN
ejpam-6813	56	5	,	,	PUNCT
ejpam-6813	56	6	k	k	PROPN
ejpam-6813	56	7	)	)	PUNCT
ejpam-6813	56	8	]	]	PUNCT
ejpam-6813	57	1	n	n	PRON
ejpam-6813	57	2	l	l	NOUN
ejpam-6813	57	3	,	,	PUNCT
ejpam-6813	57	4	k=1	k=1	X
ejpam-6813	57	5	,	,	PUNCT
ejpam-6813	57	6	w	w	PROPN
ejpam-6813	57	7	(	(	PUNCT
ejpam-6813	57	8	2	2	NUM
ejpam-6813	57	9	)	)	PUNCT
ejpam-6813	57	10	n	n	CCONJ
ejpam-6813	57	11	(	(	PUNCT
ejpam-6813	57	12	a	a	X
ejpam-6813	57	13	)	)	PUNCT
ejpam-6813	57	14	=	=	PUNCT
ejpam-6813	58	1	[	[	X
ejpam-6813	58	2	wn	wn	X
ejpam-6813	58	3	(	(	PUNCT
ejpam-6813	58	4	ai	ai	PROPN
ejpam-6813	58	5	,	,	PUNCT
ejpam-6813	58	6	j	j	PROPN
ejpam-6813	58	7	)	)	PUNCT
ejpam-6813	58	8	]	]	PUNCT
ejpam-6813	59	1	m	m	VERB
ejpam-6813	59	2	i	i	PRON
ejpam-6813	59	3	,	,	PUNCT
ejpam-6813	59	4	j=1	j=1	PROPN
ejpam-6813	59	5	,	,	PUNCT
ejpam-6813	59	6	w(1)(a	w(1)(a	NOUN
ejpam-6813	59	7	)	)	PUNCT
ejpam-6813	59	8	=	=	NOUN
ejpam-6813	60	1	[	[	X
ejpam-6813	60	2	w(gl	w(gl	NOUN
ejpam-6813	60	3	,	,	PUNCT
ejpam-6813	60	4	k	k	NOUN
ejpam-6813	60	5	)	)	PUNCT
ejpam-6813	60	6	]	]	PUNCT
ejpam-6813	60	7	n	n	PRON
ejpam-6813	60	8	l	l	NOUN
ejpam-6813	60	9	,	,	PUNCT
ejpam-6813	60	10	k=1	k=1	X
ejpam-6813	60	11	,	,	PUNCT
ejpam-6813	60	12	and	and	CCONJ
ejpam-6813	60	13	w(2)(a	w(2)(a	NOUN
ejpam-6813	60	14	)	)	PUNCT
ejpam-6813	60	15	=	=	NOUN
ejpam-6813	61	1	[	[	X
ejpam-6813	61	2	w(ai	w(ai	PROPN
ejpam-6813	61	3	,	,	PUNCT
ejpam-6813	61	4	j	j	PROPN
ejpam-6813	61	5	)	)	PUNCT
ejpam-6813	61	6	]	]	PUNCT
ejpam-6813	62	1	m	m	VERB
ejpam-6813	62	2	i	i	PRON
ejpam-6813	62	3	,	,	PUNCT
ejpam-6813	62	4	j=1	j=1	PROPN
ejpam-6813	62	5	.	.	PUNCT
ejpam-6813	63	1	these	these	DET
ejpam-6813	63	2	constructions	construction	NOUN
ejpam-6813	63	3	extend	extend	VERB
ejpam-6813	63	4	classical	classical	ADJ
ejpam-6813	63	5	numerical	numerical	ADJ
ejpam-6813	63	6	radius	radius	NOUN
ejpam-6813	63	7	concepts	concept	NOUN
ejpam-6813	63	8	to	to	PART
ejpam-6813	63	9	block	block	VERB
ejpam-6813	63	10	matrix	matrix	NOUN
ejpam-6813	63	11	frameworks	framework	NOUN
ejpam-6813	63	12	while	while	SCONJ
ejpam-6813	63	13	preserving	preserve	VERB
ejpam-6813	63	14	fundamental	fundamental	ADJ
ejpam-6813	63	15	properties	property	NOUN
ejpam-6813	63	16	and	and	CCONJ
ejpam-6813	63	17	enabling	enable	VERB
ejpam-6813	63	18	new	new	ADJ
ejpam-6813	63	19	theoretical	theoretical	ADJ
ejpam-6813	63	20	developments	development	NOUN
ejpam-6813	63	21	.	.	PUNCT
ejpam-6813	64	1	we	we	PRON
ejpam-6813	64	2	establish	establish	VERB
ejpam-6813	64	3	various	various	ADJ
ejpam-6813	64	4	properties	property	NOUN
ejpam-6813	64	5	and	and	CCONJ
ejpam-6813	64	6	inequalities	inequality	NOUN
ejpam-6813	64	7	connecting	connect	VERB
ejpam-6813	64	8	these	these	DET
ejpam-6813	64	9	novel	novel	ADJ
ejpam-6813	64	10	definitions	definition	NOUN
ejpam-6813	64	11	to	to	ADP
ejpam-6813	64	12	established	established	ADJ
ejpam-6813	64	13	matrix	matrix	NOUN
ejpam-6813	64	14	parameters	parameter	NOUN
ejpam-6813	64	15	,	,	PUNCT
ejpam-6813	64	16	thereby	thereby	ADV
ejpam-6813	64	17	generalizing	generalize	VERB
ejpam-6813	64	18	well	well	ADV
ejpam-6813	64	19	-	-	PUNCT
ejpam-6813	64	20	known	know	VERB
ejpam-6813	64	21	classical	classical	ADJ
ejpam-6813	64	22	results	result	NOUN
ejpam-6813	64	23	to	to	ADP
ejpam-6813	64	24	the	the	DET
ejpam-6813	64	25	block	block	NOUN
ejpam-6813	64	26	matrix	matrix	NOUN
ejpam-6813	64	27	setting	setting	NOUN
ejpam-6813	64	28	.	.	PUNCT
ejpam-6813	65	1	the	the	DET
ejpam-6813	65	2	key	key	ADJ
ejpam-6813	65	3	insights	insight	NOUN
ejpam-6813	65	4	of	of	ADP
ejpam-6813	65	5	our	our	PRON
ejpam-6813	65	6	approach	approach	NOUN
ejpam-6813	65	7	lie	lie	VERB
ejpam-6813	65	8	in	in	ADP
ejpam-6813	65	9	recognizing	recognize	VERB
ejpam-6813	65	10	that	that	SCONJ
ejpam-6813	65	11	while	while	SCONJ
ejpam-6813	65	12	eigenvalue	eigenvalue	ADJ
ejpam-6813	65	13	analysis	analysis	NOUN
ejpam-6813	65	14	provides	provide	VERB
ejpam-6813	65	15	spectral	spectral	ADJ
ejpam-6813	65	16	information	information	NOUN
ejpam-6813	65	17	,	,	PUNCT
ejpam-6813	65	18	the	the	DET
ejpam-6813	65	19	numerical	numerical	PROPN
ejpam-6813	65	20	radius	radius	PROPN
ejpam-6813	65	21	captures	capture	VERB
ejpam-6813	65	22	geometric	geometric	ADJ
ejpam-6813	65	23	properties	property	NOUN
ejpam-6813	65	24	of	of	ADP
ejpam-6813	65	25	the	the	DET
ejpam-6813	65	26	field	field	NOUN
ejpam-6813	65	27	of	of	ADP
ejpam-6813	65	28	values	value	NOUN
ejpam-6813	65	29	that	that	PRON
ejpam-6813	65	30	are	be	AUX
ejpam-6813	65	31	particularly	particularly	ADV
ejpam-6813	65	32	relevant	relevant	ADJ
ejpam-6813	65	33	for	for	ADP
ejpam-6813	65	34	non	non	ADJ
ejpam-6813	65	35	-	-	ADJ
ejpam-6813	65	36	normal	normal	ADJ
ejpam-6813	65	37	matrices	matrix	NOUN
ejpam-6813	65	38	and	and	CCONJ
ejpam-6813	65	39	complex	complex	ADJ
ejpam-6813	65	40	block	block	NOUN
ejpam-6813	65	41	structures	structure	NOUN
ejpam-6813	65	42	.	.	PUNCT
ejpam-6813	66	1	this	this	DET
ejpam-6813	66	2	geometric	geometric	ADJ
ejpam-6813	66	3	perspective	perspective	NOUN
ejpam-6813	66	4	complements	complement	VERB
ejpam-6813	66	5	traditional	traditional	ADJ
ejpam-6813	66	6	spectral	spectral	ADJ
ejpam-6813	66	7	methods	method	NOUN
ejpam-6813	66	8	and	and	CCONJ
ejpam-6813	66	9	provides	provide	VERB
ejpam-6813	66	10	new	new	ADJ
ejpam-6813	66	11	tools	tool	NOUN
ejpam-6813	66	12	for	for	ADP
ejpam-6813	66	13	analyzing	analyze	VERB
ejpam-6813	66	14	matrix	matrix	NOUN
ejpam-6813	66	15	behavior	behavior	NOUN
ejpam-6813	66	16	in	in	ADP
ejpam-6813	66	17	applications	application	NOUN
ejpam-6813	66	18	ranging	range	VERB
ejpam-6813	66	19	from	from	ADP
ejpam-6813	66	20	quantum	quantum	ADJ
ejpam-6813	66	21	mechanics	mechanic	NOUN
ejpam-6813	66	22	to	to	PART
ejpam-6813	66	23	control	control	VERB
ejpam-6813	66	24	theory	theory	NOUN
ejpam-6813	66	25	.	.	PUNCT
ejpam-6813	67	1	recent	recent	ADJ
ejpam-6813	67	2	advances	advance	NOUN
ejpam-6813	67	3	in	in	ADP
ejpam-6813	67	4	partial	partial	ADJ
ejpam-6813	67	5	matrix	matrix	NOUN
ejpam-6813	67	6	theory	theory	NOUN
ejpam-6813	67	7	have	have	AUX
ejpam-6813	67	8	provided	provide	VERB
ejpam-6813	67	9	additional	additional	ADJ
ejpam-6813	67	10	tools	tool	NOUN
ejpam-6813	67	11	for	for	ADP
ejpam-6813	67	12	analyzing	analyze	VERB
ejpam-6813	67	13	block	block	NOUN
ejpam-6813	67	14	matrix	matrix	NOUN
ejpam-6813	67	15	structures	structure	NOUN
ejpam-6813	67	16	.	.	PUNCT
ejpam-6813	68	1	the	the	DET
ejpam-6813	68	2	concept	concept	NOUN
ejpam-6813	68	3	of	of	ADP
ejpam-6813	68	4	partial	partial	ADJ
ejpam-6813	68	5	spectral	spectral	ADJ
ejpam-6813	68	6	radius	radius	NOUN
ejpam-6813	68	7	and	and	CCONJ
ejpam-6813	68	8	partial	partial	ADJ
ejpam-6813	68	9	matrix	matrix	NOUN
ejpam-6813	68	10	norms	norm	VERB
ejpam-6813	68	11	r.	r.	PROPN
ejpam-6813	68	12	al	al	PROPN
ejpam-6813	68	13	-	-	PROPN
ejpam-6813	68	14	naimi	naimi	PROPN
ejpam-6813	68	15	et	et	PROPN
ejpam-6813	68	16	al	al	PROPN
ejpam-6813	68	17	.	.	PUNCT
ejpam-6813	68	18	/	/	SYM
ejpam-6813	68	19	eur	eur	PROPN
ejpam-6813	68	20	.	.	PUNCT
ejpam-6813	69	1	j.	j.	PROPN
ejpam-6813	69	2	pure	pure	PROPN
ejpam-6813	69	3	appl	appl	PROPN
ejpam-6813	69	4	.	.	PROPN
ejpam-6813	69	5	math	math	PROPN
ejpam-6813	69	6	,	,	PUNCT
ejpam-6813	69	7	18	18	NUM
ejpam-6813	69	8	(	(	PUNCT
ejpam-6813	69	9	4	4	NUM
ejpam-6813	69	10	)	)	PUNCT
ejpam-6813	69	11	(	(	PUNCT
ejpam-6813	69	12	2025	2025	NUM
ejpam-6813	69	13	)	)	PUNCT
ejpam-6813	69	14	,	,	PUNCT
ejpam-6813	69	15	6813	6813	NUM
ejpam-6813	69	16	4	4	NUM
ejpam-6813	69	17	of	of	ADP
ejpam-6813	69	18	15	15	NUM
ejpam-6813	69	19	introduced	introduce	VERB
ejpam-6813	69	20	in	in	ADP
ejpam-6813	69	21	[	[	X
ejpam-6813	69	22	14	14	NUM
ejpam-6813	69	23	]	]	PUNCT
ejpam-6813	69	24	offers	offer	VERB
ejpam-6813	69	25	complementary	complementary	ADJ
ejpam-6813	69	26	perspectives	perspective	NOUN
ejpam-6813	69	27	that	that	PRON
ejpam-6813	69	28	parallel	parallel	VERB
ejpam-6813	69	29	our	our	PRON
ejpam-6813	69	30	numerical	numerical	ADJ
ejpam-6813	69	31	radius	radius	NOUN
ejpam-6813	69	32	approach	approach	NOUN
ejpam-6813	69	33	.	.	PUNCT
ejpam-6813	70	1	furthermore	furthermore	ADV
ejpam-6813	70	2	,	,	PUNCT
ejpam-6813	70	3	the	the	DET
ejpam-6813	70	4	generalized	generalized	ADJ
ejpam-6813	70	5	p	p	PROPN
ejpam-6813	70	6	-	-	PUNCT
ejpam-6813	70	7	numerical	numerical	ADJ
ejpam-6813	70	8	radius	radius	NOUN
ejpam-6813	70	9	framework	framework	NOUN
ejpam-6813	70	10	developed	develop	VERB
ejpam-6813	70	11	in	in	ADP
ejpam-6813	70	12	[	[	X
ejpam-6813	70	13	15	15	NUM
ejpam-6813	70	14	]	]	PUNCT
ejpam-6813	70	15	extends	extend	VERB
ejpam-6813	70	16	classical	classical	ADJ
ejpam-6813	70	17	numerical	numerical	ADJ
ejpam-6813	70	18	radius	radius	NOUN
ejpam-6813	70	19	concepts	concept	NOUN
ejpam-6813	70	20	to	to	ADP
ejpam-6813	70	21	more	more	ADJ
ejpam-6813	70	22	general	general	ADJ
ejpam-6813	70	23	operator	operator	NOUN
ejpam-6813	70	24	settings	setting	NOUN
ejpam-6813	70	25	,	,	PUNCT
ejpam-6813	70	26	providing	provide	VERB
ejpam-6813	70	27	theoretical	theoretical	ADJ
ejpam-6813	70	28	foundations	foundation	NOUN
ejpam-6813	70	29	that	that	PRON
ejpam-6813	70	30	motivate	motivate	VERB
ejpam-6813	70	31	our	our	PRON
ejpam-6813	70	32	block	block	NOUN
ejpam-6813	70	33	matrix	matrix	NOUN
ejpam-6813	70	34	constructions	construction	NOUN
ejpam-6813	70	35	.	.	PUNCT
ejpam-6813	71	1	2	2	X
ejpam-6813	71	2	.	.	X
ejpam-6813	71	3	main	main	ADJ
ejpam-6813	71	4	results	result	NOUN
ejpam-6813	71	5	and	and	CCONJ
ejpam-6813	71	6	fundamental	fundamental	ADJ
ejpam-6813	71	7	properties	property	NOUN
ejpam-6813	71	8	we	we	PRON
ejpam-6813	71	9	introduce	introduce	VERB
ejpam-6813	71	10	our	our	PRON
ejpam-6813	71	11	primary	primary	ADJ
ejpam-6813	71	12	definition	definition	NOUN
ejpam-6813	71	13	concerning	concern	VERB
ejpam-6813	71	14	partial	partial	ADJ
ejpam-6813	71	15	matrix	matrix	NOUN
ejpam-6813	71	16	generalized	generalize	VERB
ejpam-6813	71	17	numerical	numerical	ADJ
ejpam-6813	71	18	radius	radius	NOUN
ejpam-6813	71	19	.	.	PUNCT
ejpam-6813	72	1	definition	definition	NOUN
ejpam-6813	72	2	1	1	NUM
ejpam-6813	72	3	.	.	PUNCT
ejpam-6813	73	1	let	let	VERB
ejpam-6813	73	2	a	a	PRON
ejpam-6813	73	3	=	=	SYM
ejpam-6813	74	1	[	[	X
ejpam-6813	74	2	ai	ai	NOUN
ejpam-6813	74	3	,	,	PUNCT
ejpam-6813	74	4	j	j	PROPN
ejpam-6813	74	5	]	]	PUNCT
ejpam-6813	74	6	∈	∈	PROPN
ejpam-6813	74	7	mm(mn	mm(mn	PROPN
ejpam-6813	74	8	)	)	PUNCT
ejpam-6813	74	9	.	.	PUNCT
ejpam-6813	75	1	we	we	PRON
ejpam-6813	75	2	define	define	VERB
ejpam-6813	75	3	w	w	ADP
ejpam-6813	75	4	(	(	PUNCT
ejpam-6813	75	5	1	1	NUM
ejpam-6813	75	6	)	)	PUNCT
ejpam-6813	75	7	n	n	CCONJ
ejpam-6813	75	8	(	(	PUNCT
ejpam-6813	75	9	a	a	X
ejpam-6813	75	10	)	)	PUNCT
ejpam-6813	75	11	=	=	PUNCT
ejpam-6813	76	1	[	[	X
ejpam-6813	76	2	wn	wn	X
ejpam-6813	76	3	(	(	PUNCT
ejpam-6813	76	4	gl	gl	PROPN
ejpam-6813	76	5	,	,	PUNCT
ejpam-6813	76	6	k	k	PROPN
ejpam-6813	76	7	)	)	PUNCT
ejpam-6813	76	8	]	]	PUNCT
ejpam-6813	77	1	n	n	PRON
ejpam-6813	77	2	l	l	NOUN
ejpam-6813	77	3	,	,	PUNCT
ejpam-6813	77	4	k=1	k=1	PROPN
ejpam-6813	77	5	and	and	CCONJ
ejpam-6813	77	6	w	w	PROPN
ejpam-6813	77	7	(	(	PUNCT
ejpam-6813	77	8	2	2	NUM
ejpam-6813	77	9	)	)	PUNCT
ejpam-6813	77	10	n	n	CCONJ
ejpam-6813	77	11	(	(	PUNCT
ejpam-6813	77	12	a	a	X
ejpam-6813	77	13	)	)	PUNCT
ejpam-6813	78	1	=	=	PUNCT
ejpam-6813	79	1	[	[	X
ejpam-6813	79	2	wn	wn	X
ejpam-6813	79	3	(	(	PUNCT
ejpam-6813	79	4	ai	ai	PROPN
ejpam-6813	79	5	,	,	PUNCT
ejpam-6813	79	6	j	j	PROPN
ejpam-6813	79	7	)	)	PUNCT
ejpam-6813	79	8	]	]	PUNCT
ejpam-6813	80	1	m	m	VERB
ejpam-6813	80	2	i	i	PRON
ejpam-6813	80	3	,	,	PUNCT
ejpam-6813	80	4	j=1	j=1	PROPN
ejpam-6813	80	5	.	.	PUNCT
ejpam-6813	81	1	by	by	ADP
ejpam-6813	81	2	specializing	specialize	VERB
ejpam-6813	81	3	definition	definition	NOUN
ejpam-6813	81	4	1	1	NUM
ejpam-6813	81	5	with	with	ADP
ejpam-6813	81	6	n(a	n(a	NOUN
ejpam-6813	81	7	)	)	PUNCT
ejpam-6813	81	8	=	=	PUNCT
ejpam-6813	81	9	‖a‖	‖a‖	PROPN
ejpam-6813	81	10	,	,	PUNCT
ejpam-6813	81	11	we	we	PRON
ejpam-6813	81	12	obtain	obtain	VERB
ejpam-6813	81	13	the	the	DET
ejpam-6813	81	14	partial	partial	ADJ
ejpam-6813	81	15	matrix	matrix	NOUN
ejpam-6813	81	16	of	of	ADP
ejpam-6813	81	17	numerical	numerical	ADJ
ejpam-6813	81	18	radius	radius	PROPN
ejpam-6813	81	19	.	.	PUNCT
ejpam-6813	82	1	definition	definition	NOUN
ejpam-6813	82	2	2	2	NUM
ejpam-6813	82	3	.	.	PUNCT
ejpam-6813	83	1	let	let	VERB
ejpam-6813	83	2	a	a	PRON
ejpam-6813	83	3	=	=	SYM
ejpam-6813	84	1	[	[	X
ejpam-6813	84	2	ai	ai	NOUN
ejpam-6813	84	3	,	,	PUNCT
ejpam-6813	84	4	j	j	PROPN
ejpam-6813	84	5	]	]	PUNCT
ejpam-6813	84	6	∈	∈	PROPN
ejpam-6813	84	7	mm(mn	mm(mn	PROPN
ejpam-6813	84	8	)	)	PUNCT
ejpam-6813	84	9	.	.	PUNCT
ejpam-6813	85	1	we	we	PRON
ejpam-6813	85	2	define	define	VERB
ejpam-6813	85	3	w(1)(a	w(1)(a	NOUN
ejpam-6813	85	4	)	)	PUNCT
ejpam-6813	85	5	=	=	NOUN
ejpam-6813	86	1	[	[	X
ejpam-6813	86	2	w(gl	w(gl	NOUN
ejpam-6813	86	3	,	,	PUNCT
ejpam-6813	86	4	k	k	NOUN
ejpam-6813	86	5	)	)	PUNCT
ejpam-6813	86	6	]	]	PUNCT
ejpam-6813	86	7	n	n	PRON
ejpam-6813	86	8	l	l	NOUN
ejpam-6813	86	9	,	,	PUNCT
ejpam-6813	86	10	k=1	k=1	PROPN
ejpam-6813	86	11	and	and	CCONJ
ejpam-6813	86	12	w(2)(a	w(2)(a	NOUN
ejpam-6813	86	13	)	)	PUNCT
ejpam-6813	87	1	=	=	VERB
ejpam-6813	88	1	[	[	X
ejpam-6813	88	2	w(ai	w(ai	PROPN
ejpam-6813	88	3	,	,	PUNCT
ejpam-6813	88	4	j	j	PROPN
ejpam-6813	88	5	)	)	PUNCT
ejpam-6813	88	6	]	]	PUNCT
ejpam-6813	89	1	m	m	VERB
ejpam-6813	89	2	i	i	PRON
ejpam-6813	89	3	,	,	PUNCT
ejpam-6813	89	4	j=1	j=1	PROPN
ejpam-6813	89	5	.	.	PUNCT
ejpam-6813	90	1	these	these	DET
ejpam-6813	90	2	definitions	definition	NOUN
ejpam-6813	90	3	extend	extend	VERB
ejpam-6813	90	4	the	the	DET
ejpam-6813	90	5	classical	classical	ADJ
ejpam-6813	90	6	numerical	numerical	ADJ
ejpam-6813	90	7	radius	radius	NOUN
ejpam-6813	90	8	to	to	PART
ejpam-6813	90	9	block	block	NOUN
ejpam-6813	90	10	matrix	matrix	NOUN
ejpam-6813	90	11	structures	structure	NOUN
ejpam-6813	90	12	in	in	ADP
ejpam-6813	90	13	a	a	DET
ejpam-6813	90	14	natural	natural	ADJ
ejpam-6813	90	15	way	way	NOUN
ejpam-6813	90	16	.	.	PUNCT
ejpam-6813	91	1	the	the	DET
ejpam-6813	91	2	first	first	ADJ
ejpam-6813	91	3	partial	partial	ADJ
ejpam-6813	91	4	matrix	matrix	NOUN
ejpam-6813	91	5	w(1)(a	w(1)(a	NOUN
ejpam-6813	91	6	)	)	PUNCT
ejpam-6813	91	7	captures	capture	VERB
ejpam-6813	91	8	the	the	DET
ejpam-6813	91	9	numerical	numerical	ADJ
ejpam-6813	91	10	radius	radius	NOUN
ejpam-6813	91	11	properties	property	NOUN
ejpam-6813	91	12	when	when	SCONJ
ejpam-6813	91	13	we	we	PRON
ejpam-6813	91	14	view	view	VERB
ejpam-6813	91	15	the	the	DET
ejpam-6813	91	16	matrix	matrix	NOUN
ejpam-6813	91	17	through	through	ADP
ejpam-6813	91	18	its	its	PRON
ejpam-6813	91	19	column	column	NOUN
ejpam-6813	91	20	-	-	PUNCT
ejpam-6813	91	21	block	block	NOUN
ejpam-6813	91	22	structure	structure	NOUN
ejpam-6813	91	23	,	,	PUNCT
ejpam-6813	91	24	while	while	SCONJ
ejpam-6813	91	25	the	the	DET
ejpam-6813	91	26	second	second	ADJ
ejpam-6813	91	27	partial	partial	ADJ
ejpam-6813	91	28	matrix	matrix	NOUN
ejpam-6813	91	29	w(2)(a	w(2)(a	NOUN
ejpam-6813	91	30	)	)	PUNCT
ejpam-6813	91	31	emphasizes	emphasize	VERB
ejpam-6813	91	32	the	the	DET
ejpam-6813	91	33	row	row	NOUN
ejpam-6813	91	34	-	-	PUNCT
ejpam-6813	91	35	block	block	NOUN
ejpam-6813	91	36	perspective	perspective	NOUN
ejpam-6813	91	37	.	.	PUNCT
ejpam-6813	92	1	the	the	DET
ejpam-6813	92	2	partial	partial	ADJ
ejpam-6813	92	3	numerical	numerical	ADJ
ejpam-6813	92	4	radius	radius	PROPN
ejpam-6813	92	5	constructions	construction	NOUN
ejpam-6813	92	6	introduced	introduce	VERB
ejpam-6813	92	7	here	here	ADV
ejpam-6813	92	8	naturally	naturally	ADV
ejpam-6813	92	9	extend	extend	VERB
ejpam-6813	92	10	to	to	ADP
ejpam-6813	92	11	the	the	DET
ejpam-6813	92	12	p	p	PROPN
ejpam-6813	92	13	-	-	PUNCT
ejpam-6813	92	14	numerical	numerical	ADJ
ejpam-6813	92	15	radius	radius	NOUN
ejpam-6813	92	16	setting	setting	NOUN
ejpam-6813	92	17	studied	study	VERB
ejpam-6813	92	18	in	in	ADP
ejpam-6813	92	19	[	[	X
ejpam-6813	92	20	15	15	NUM
ejpam-6813	92	21	]	]	PUNCT
ejpam-6813	92	22	.	.	PUNCT
ejpam-6813	93	1	while	while	SCONJ
ejpam-6813	93	2	[	[	X
ejpam-6813	93	3	15	15	NUM
ejpam-6813	93	4	]	]	PUNCT
ejpam-6813	93	5	establishes	establish	VERB
ejpam-6813	93	6	p	p	X
ejpam-6813	93	7	-	-	PUNCT
ejpam-6813	93	8	numerical	numerical	ADJ
ejpam-6813	93	9	radius	radius	NOUN
ejpam-6813	93	10	inequalities	inequality	NOUN
ejpam-6813	93	11	for	for	ADP
ejpam-6813	93	12	general	general	ADJ
ejpam-6813	93	13	operators	operator	NOUN
ejpam-6813	93	14	on	on	ADP
ejpam-6813	93	15	hilbert	hilbert	PROPN
ejpam-6813	93	16	spaces	space	NOUN
ejpam-6813	93	17	,	,	PUNCT
ejpam-6813	93	18	our	our	PRON
ejpam-6813	93	19	framework	framework	NOUN
ejpam-6813	93	20	specializes	specialize	VERB
ejpam-6813	93	21	these	these	DET
ejpam-6813	93	22	concepts	concept	NOUN
ejpam-6813	93	23	to	to	PART
ejpam-6813	93	24	block	block	VERB
ejpam-6813	93	25	matrix	matrix	NOUN
ejpam-6813	93	26	structures	structure	NOUN
ejpam-6813	93	27	where	where	SCONJ
ejpam-6813	93	28	each	each	DET
ejpam-6813	93	29	block	block	NOUN
ejpam-6813	93	30	admits	admit	VERB
ejpam-6813	93	31	finite	finite	ADJ
ejpam-6813	93	32	-	-	ADJ
ejpam-6813	93	33	dimensional	dimensional	ADJ
ejpam-6813	93	34	matrix	matrix	NOUN
ejpam-6813	93	35	representation	representation	NOUN
ejpam-6813	93	36	.	.	PUNCT
ejpam-6813	94	1	the	the	DET
ejpam-6813	94	2	relationship	relationship	NOUN
ejpam-6813	94	3	between	between	ADP
ejpam-6813	94	4	our	our	PRON
ejpam-6813	94	5	partial	partial	ADJ
ejpam-6813	94	6	numerical	numerical	ADJ
ejpam-6813	94	7	radius	radius	NOUN
ejpam-6813	94	8	matrices	matrix	NOUN
ejpam-6813	94	9	and	and	CCONJ
ejpam-6813	94	10	the	the	DET
ejpam-6813	94	11	partial	partial	ADJ
ejpam-6813	94	12	spectral	spectral	ADJ
ejpam-6813	94	13	radius	radius	NOUN
ejpam-6813	94	14	constructions	construction	NOUN
ejpam-6813	94	15	in	in	ADP
ejpam-6813	94	16	cite14	cite14	NOUN
ejpam-6813	94	17	reveals	reveal	VERB
ejpam-6813	94	18	deeper	deep	ADJ
ejpam-6813	94	19	structural	structural	ADJ
ejpam-6813	94	20	connections	connection	NOUN
ejpam-6813	94	21	between	between	ADP
ejpam-6813	94	22	geometric	geometric	ADJ
ejpam-6813	94	23	and	and	CCONJ
ejpam-6813	94	24	spectral	spectral	ADJ
ejpam-6813	94	25	properties	property	NOUN
ejpam-6813	94	26	of	of	ADP
ejpam-6813	94	27	block	block	NOUN
ejpam-6813	94	28	matrices	matrix	NOUN
ejpam-6813	94	29	,	,	PUNCT
ejpam-6813	94	30	suggesting	suggest	VERB
ejpam-6813	94	31	directions	direction	NOUN
ejpam-6813	94	32	for	for	ADP
ejpam-6813	94	33	unified	unified	ADJ
ejpam-6813	94	34	treatments	treatment	NOUN
ejpam-6813	94	35	in	in	ADP
ejpam-6813	94	36	future	future	ADJ
ejpam-6813	94	37	work	work	NOUN
ejpam-6813	94	38	.	.	PUNCT
ejpam-6813	95	1	remark	remark	NOUN
ejpam-6813	95	2	1	1	NUM
ejpam-6813	95	3	.	.	PUNCT
ejpam-6813	96	1	for	for	ADP
ejpam-6813	96	2	any	any	DET
ejpam-6813	96	3	matrix	matrix	NOUN
ejpam-6813	96	4	a	a	PRON
ejpam-6813	96	5	=	=	PUNCT
ejpam-6813	97	1	[	[	X
ejpam-6813	97	2	ai	ai	NOUN
ejpam-6813	97	3	,	,	PUNCT
ejpam-6813	97	4	j	j	PROPN
ejpam-6813	97	5	]	]	PUNCT
ejpam-6813	97	6	∈	∈	PROPN
ejpam-6813	97	7	mm(mn	mm(mn	PROPN
ejpam-6813	97	8	)	)	PUNCT
ejpam-6813	97	9	,	,	PUNCT
ejpam-6813	97	10	the	the	DET
ejpam-6813	97	11	following	follow	VERB
ejpam-6813	97	12	properties	property	NOUN
ejpam-6813	97	13	hold	hold	VERB
ejpam-6813	97	14	:	:	PUNCT
ejpam-6813	97	15	1	1	X
ejpam-6813	97	16	.	.	X
ejpam-6813	97	17	w	w	NOUN
ejpam-6813	97	18	(	(	PUNCT
ejpam-6813	97	19	1	1	NUM
ejpam-6813	97	20	)	)	PUNCT
ejpam-6813	97	21	n	n	CCONJ
ejpam-6813	97	22	(	(	PUNCT
ejpam-6813	97	23	a	a	X
ejpam-6813	97	24	)	)	PUNCT
ejpam-6813	97	25	∈	∈	NOUN
ejpam-6813	97	26	mn(c	mn(c	X
ejpam-6813	97	27	)	)	PUNCT
ejpam-6813	97	28	and	and	CCONJ
ejpam-6813	97	29	w	w	PROPN
ejpam-6813	97	30	(	(	PUNCT
ejpam-6813	97	31	2	2	NUM
ejpam-6813	97	32	)	)	PUNCT
ejpam-6813	97	33	n	n	CCONJ
ejpam-6813	97	34	(	(	PUNCT
ejpam-6813	97	35	a	a	X
ejpam-6813	97	36	)	)	PUNCT
ejpam-6813	97	37	∈	∈	NOUN
ejpam-6813	97	38	mm(c	mm(c	NOUN
ejpam-6813	97	39	)	)	PUNCT
ejpam-6813	97	40	.	.	PUNCT
ejpam-6813	98	1	2	2	X
ejpam-6813	98	2	.	.	X
ejpam-6813	98	3	all	all	DET
ejpam-6813	98	4	entries	entry	NOUN
ejpam-6813	98	5	of	of	ADP
ejpam-6813	98	6	w(1	w(1	PROPN
ejpam-6813	98	7	)	)	PUNCT
ejpam-6813	98	8	n	n	CCONJ
ejpam-6813	98	9	(	(	PUNCT
ejpam-6813	98	10	a	a	NOUN
ejpam-6813	98	11	)	)	PUNCT
ejpam-6813	98	12	and	and	CCONJ
ejpam-6813	98	13	w	w	PROPN
ejpam-6813	98	14	(	(	PUNCT
ejpam-6813	98	15	2	2	NUM
ejpam-6813	98	16	)	)	PUNCT
ejpam-6813	98	17	n	n	CCONJ
ejpam-6813	98	18	(	(	PUNCT
ejpam-6813	98	19	a	a	X
ejpam-6813	98	20	)	)	PUNCT
ejpam-6813	98	21	are	be	AUX
ejpam-6813	98	22	positive	positive	ADJ
ejpam-6813	98	23	.	.	PUNCT
ejpam-6813	99	1	3	3	X
ejpam-6813	99	2	.	.	X
ejpam-6813	99	3	the	the	DET
ejpam-6813	99	4	duality	duality	NOUN
ejpam-6813	99	5	relations	relation	NOUN
ejpam-6813	99	6	w	w	PROPN
ejpam-6813	99	7	(	(	PUNCT
ejpam-6813	99	8	2	2	NUM
ejpam-6813	99	9	)	)	PUNCT
ejpam-6813	99	10	n	n	CCONJ
ejpam-6813	99	11	(	(	PUNCT
ejpam-6813	99	12	a	a	X
ejpam-6813	99	13	)	)	PUNCT
ejpam-6813	99	14	=	=	SYM
ejpam-6813	99	15	w	w	PROPN
ejpam-6813	99	16	(	(	PUNCT
ejpam-6813	99	17	1	1	NUM
ejpam-6813	99	18	)	)	PUNCT
ejpam-6813	99	19	n	n	PROPN
ejpam-6813	99	20	(	(	PUNCT
ejpam-6813	99	21	ã	ã	PROPN
ejpam-6813	99	22	)	)	PUNCT
ejpam-6813	99	23	and	and	CCONJ
ejpam-6813	99	24	w	w	PROPN
ejpam-6813	99	25	(	(	PUNCT
ejpam-6813	99	26	1	1	NUM
ejpam-6813	99	27	)	)	PUNCT
ejpam-6813	99	28	n	n	CCONJ
ejpam-6813	99	29	(	(	PUNCT
ejpam-6813	99	30	a	a	X
ejpam-6813	99	31	)	)	PUNCT
ejpam-6813	99	32	=	=	SYM
ejpam-6813	99	33	w	w	PROPN
ejpam-6813	99	34	(	(	PUNCT
ejpam-6813	99	35	2	2	NUM
ejpam-6813	99	36	)	)	PUNCT
ejpam-6813	99	37	n	n	PROPN
ejpam-6813	99	38	(	(	PUNCT
ejpam-6813	99	39	ã	ã	PROPN
ejpam-6813	99	40	)	)	PUNCT
ejpam-6813	99	41	are	be	AUX
ejpam-6813	99	42	satisfied	satisfied	ADJ
ejpam-6813	99	43	.	.	PUNCT
ejpam-6813	100	1	lemma	lemma	PROPN
ejpam-6813	100	2	1	1	NUM
ejpam-6813	100	3	.	.	PUNCT
ejpam-6813	101	1	for	for	ADP
ejpam-6813	101	2	any	any	DET
ejpam-6813	101	3	matrix	matrix	NOUN
ejpam-6813	101	4	a	a	PRON
ejpam-6813	101	5	=	=	PUNCT
ejpam-6813	102	1	[	[	X
ejpam-6813	102	2	ai	ai	NOUN
ejpam-6813	102	3	,	,	PUNCT
ejpam-6813	102	4	j	j	PROPN
ejpam-6813	102	5	]	]	PUNCT
ejpam-6813	102	6	∈	∈	PROPN
ejpam-6813	102	7	mm(mn	mm(mn	PROPN
ejpam-6813	102	8	)	)	PUNCT
ejpam-6813	102	9	,	,	PUNCT
ejpam-6813	102	10	we	we	PRON
ejpam-6813	102	11	have	have	VERB
ejpam-6813	102	12	α̃a	α̃a	PROPN
ejpam-6813	102	13	=	=	SYM
ejpam-6813	102	14	αã.	αã.	NOUN
ejpam-6813	102	15	proof	proof	NOUN
ejpam-6813	102	16	.	.	PUNCT
ejpam-6813	103	1	this	this	PRON
ejpam-6813	103	2	follows	follow	VERB
ejpam-6813	103	3	directly	directly	ADV
ejpam-6813	103	4	from	from	ADP
ejpam-6813	103	5	the	the	DET
ejpam-6813	103	6	definition	definition	NOUN
ejpam-6813	103	7	of	of	ADP
ejpam-6813	103	8	the	the	DET
ejpam-6813	103	9	tilde	tilde	ADJ
ejpam-6813	103	10	transformation	transformation	NOUN
ejpam-6813	103	11	and	and	CCONJ
ejpam-6813	103	12	scalar	scalar	ADJ
ejpam-6813	103	13	multiplication	multiplication	NOUN
ejpam-6813	103	14	.	.	PUNCT
ejpam-6813	104	1	for	for	ADP
ejpam-6813	104	2	αa	αa	NOUN
ejpam-6813	104	3	=	=	SYM
ejpam-6813	104	4	[	[	X
ejpam-6813	104	5	αai	αai	PROPN
ejpam-6813	104	6	,	,	PUNCT
ejpam-6813	104	7	j	j	PROPN
ejpam-6813	104	8	]	]	PUNCT
ejpam-6813	104	9	,	,	PUNCT
ejpam-6813	104	10	we	we	PRON
ejpam-6813	104	11	have	have	VERB
ejpam-6813	104	12	α̃a	α̃a	PROPN
ejpam-6813	104	13	=	=	PUNCT
ejpam-6813	105	1	[	[	X
ejpam-6813	105	2	[	[	X
ejpam-6813	105	3	αai	αai	ADJ
ejpam-6813	105	4	,	,	PUNCT
ejpam-6813	105	5	jl	jl	NOUN
ejpam-6813	105	6	,	,	PUNCT
ejpam-6813	105	7	k	k	X
ejpam-6813	105	8	]	]	X
ejpam-6813	105	9	m	m	VERB
ejpam-6813	105	10	i	i	PRON
ejpam-6813	105	11	,	,	PUNCT
ejpam-6813	105	12	j=1	j=1	X
ejpam-6813	105	13	]	]	X
ejpam-6813	105	14	n	n	PRON
ejpam-6813	105	15	l	l	NOUN
ejpam-6813	105	16	,	,	PUNCT
ejpam-6813	105	17	k=1	k=1	PROPN
ejpam-6813	105	18	=	=	SYM
ejpam-6813	105	19	α[[ai	α[[ai	PROPN
ejpam-6813	105	20	,	,	PUNCT
ejpam-6813	105	21	jl	jl	NOUN
ejpam-6813	105	22	,	,	PUNCT
ejpam-6813	105	23	k	k	X
ejpam-6813	105	24	]	]	X
ejpam-6813	105	25	m	m	VERB
ejpam-6813	105	26	i	i	PRON
ejpam-6813	105	27	,	,	PUNCT
ejpam-6813	105	28	j=1	j=1	X
ejpam-6813	105	29	]	]	X
ejpam-6813	105	30	n	n	PRON
ejpam-6813	105	31	l	l	NOUN
ejpam-6813	105	32	,	,	PUNCT
ejpam-6813	105	33	k=1	k=1	PUNCT
ejpam-6813	106	1	=	=	PUNCT
ejpam-6813	106	2	αã.	αã.	X
ejpam-6813	106	3	the	the	DET
ejpam-6813	106	4	following	follow	VERB
ejpam-6813	106	5	theorem	theorem	NOUN
ejpam-6813	106	6	establishes	establish	VERB
ejpam-6813	106	7	the	the	DET
ejpam-6813	106	8	homogeneity	homogeneity	NOUN
ejpam-6813	106	9	property	property	NOUN
ejpam-6813	106	10	,	,	PUNCT
ejpam-6813	106	11	which	which	PRON
ejpam-6813	106	12	is	be	AUX
ejpam-6813	106	13	fundamental	fundamental	ADJ
ejpam-6813	106	14	for	for	ADP
ejpam-6813	106	15	any	any	DET
ejpam-6813	106	16	meaningful	meaningful	ADJ
ejpam-6813	106	17	extension	extension	NOUN
ejpam-6813	106	18	of	of	ADP
ejpam-6813	106	19	the	the	DET
ejpam-6813	106	20	numerical	numerical	PROPN
ejpam-6813	106	21	radius	radius	PROPN
ejpam-6813	106	22	concept	concept	NOUN
ejpam-6813	106	23	.	.	PUNCT
ejpam-6813	107	1	r.	r.	PROPN
ejpam-6813	107	2	al	al	PROPN
ejpam-6813	107	3	-	-	PROPN
ejpam-6813	107	4	naimi	naimi	PROPN
ejpam-6813	107	5	et	et	PROPN
ejpam-6813	107	6	al	al	PROPN
ejpam-6813	107	7	.	.	PUNCT
ejpam-6813	107	8	/	/	SYM
ejpam-6813	107	9	eur	eur	PROPN
ejpam-6813	107	10	.	.	PUNCT
ejpam-6813	108	1	j.	j.	PROPN
ejpam-6813	108	2	pure	pure	PROPN
ejpam-6813	108	3	appl	appl	PROPN
ejpam-6813	108	4	.	.	PROPN
ejpam-6813	108	5	math	math	PROPN
ejpam-6813	108	6	,	,	PUNCT
ejpam-6813	108	7	18	18	NUM
ejpam-6813	108	8	(	(	PUNCT
ejpam-6813	108	9	4	4	NUM
ejpam-6813	108	10	)	)	PUNCT
ejpam-6813	108	11	(	(	PUNCT
ejpam-6813	108	12	2025	2025	NUM
ejpam-6813	108	13	)	)	PUNCT
ejpam-6813	108	14	,	,	PUNCT
ejpam-6813	108	15	6813	6813	NUM
ejpam-6813	108	16	5	5	NUM
ejpam-6813	108	17	of	of	ADP
ejpam-6813	108	18	15	15	NUM
ejpam-6813	108	19	theorem	theorem	NOUN
ejpam-6813	108	20	1	1	NUM
ejpam-6813	108	21	.	.	PUNCT
ejpam-6813	108	22	given	give	VERB
ejpam-6813	108	23	a	a	PRON
ejpam-6813	108	24	=	=	PRON
ejpam-6813	109	1	[	[	X
ejpam-6813	109	2	ai	ai	NOUN
ejpam-6813	109	3	,	,	PUNCT
ejpam-6813	109	4	j	j	PROPN
ejpam-6813	109	5	]	]	PUNCT
ejpam-6813	109	6	∈	∈	PROPN
ejpam-6813	109	7	mm(mn	mm(mn	PROPN
ejpam-6813	109	8	)	)	PUNCT
ejpam-6813	109	9	,	,	PUNCT
ejpam-6813	109	10	the	the	DET
ejpam-6813	109	11	homogeneity	homogeneity	NOUN
ejpam-6813	109	12	properties	property	NOUN
ejpam-6813	109	13	w	w	X
ejpam-6813	109	14	(	(	PUNCT
ejpam-6813	109	15	1	1	NUM
ejpam-6813	109	16	)	)	PUNCT
ejpam-6813	109	17	n	n	PROPN
ejpam-6813	109	18	(	(	PUNCT
ejpam-6813	109	19	αa	αa	NOUN
ejpam-6813	109	20	)	)	PUNCT
ejpam-6813	109	21	=	=	PUNCT
ejpam-6813	109	22	|α|w(1	|α|w(1	X
ejpam-6813	109	23	)	)	PUNCT
ejpam-6813	110	1	n	n	CCONJ
ejpam-6813	110	2	(	(	PUNCT
ejpam-6813	110	3	a	a	NOUN
ejpam-6813	110	4	)	)	PUNCT
ejpam-6813	110	5	and	and	CCONJ
ejpam-6813	110	6	w	w	PROPN
ejpam-6813	110	7	(	(	PUNCT
ejpam-6813	110	8	2	2	NUM
ejpam-6813	110	9	)	)	PUNCT
ejpam-6813	110	10	n	n	CCONJ
ejpam-6813	110	11	(	(	PUNCT
ejpam-6813	110	12	αa	αa	NOUN
ejpam-6813	110	13	)	)	PUNCT
ejpam-6813	110	14	=	=	SYM
ejpam-6813	110	15	|α|w(2	|α|w(2	NOUN
ejpam-6813	110	16	)	)	PUNCT
ejpam-6813	110	17	n	n	CCONJ
ejpam-6813	110	18	(	(	PUNCT
ejpam-6813	110	19	a	a	PRON
ejpam-6813	110	20	)	)	PUNCT
ejpam-6813	110	21	hold	hold	NOUN
ejpam-6813	110	22	.	.	PUNCT
ejpam-6813	111	1	proof	proof	NOUN
ejpam-6813	111	2	.	.	PUNCT
ejpam-6813	112	1	consider	consider	VERB
ejpam-6813	112	2	a	a	PRON
ejpam-6813	112	3	=	=	SYM
ejpam-6813	113	1	[	[	X
ejpam-6813	113	2	ai	ai	NOUN
ejpam-6813	113	3	,	,	PUNCT
ejpam-6813	113	4	j	j	PROPN
ejpam-6813	113	5	]	]	PUNCT
ejpam-6813	113	6	∈	∈	PROPN
ejpam-6813	113	7	mm(mn	mm(mn	PROPN
ejpam-6813	113	8	)	)	PUNCT
ejpam-6813	113	9	.	.	PUNCT
ejpam-6813	114	1	then	then	ADV
ejpam-6813	114	2	w	w	X
ejpam-6813	114	3	(	(	PUNCT
ejpam-6813	114	4	2	2	NUM
ejpam-6813	114	5	)	)	PUNCT
ejpam-6813	114	6	n	n	CCONJ
ejpam-6813	114	7	(	(	PUNCT
ejpam-6813	114	8	αa	αa	NOUN
ejpam-6813	114	9	)	)	PUNCT
ejpam-6813	114	10	=	=	PUNCT
ejpam-6813	115	1	[	[	X
ejpam-6813	115	2	wn	wn	X
ejpam-6813	115	3	(	(	PUNCT
ejpam-6813	115	4	αai	αai	PROPN
ejpam-6813	115	5	,	,	PUNCT
ejpam-6813	115	6	j	j	PROPN
ejpam-6813	115	7	)	)	PUNCT
ejpam-6813	115	8	]	]	PUNCT
ejpam-6813	116	1	=	=	PUNCT
ejpam-6813	117	1	[	[	X
ejpam-6813	117	2	|α|wn	|α|wn	ADJ
ejpam-6813	117	3	(	(	PUNCT
ejpam-6813	117	4	ai	ai	PROPN
ejpam-6813	117	5	,	,	PUNCT
ejpam-6813	117	6	j	j	PROPN
ejpam-6813	117	7	)	)	PUNCT
ejpam-6813	117	8	]	]	PUNCT
ejpam-6813	118	1	=	=	PUNCT
ejpam-6813	118	2	|α|[wn	|α|[wn	PROPN
ejpam-6813	118	3	(	(	PUNCT
ejpam-6813	118	4	ai	ai	PROPN
ejpam-6813	118	5	,	,	PUNCT
ejpam-6813	118	6	j	j	PROPN
ejpam-6813	118	7	)	)	PUNCT
ejpam-6813	118	8	]	]	PUNCT
ejpam-6813	118	9	=	=	SYM
ejpam-6813	118	10	|α|w(2	|α|w(2	NOUN
ejpam-6813	118	11	)	)	PUNCT
ejpam-6813	118	12	n	n	CCONJ
ejpam-6813	118	13	(	(	PUNCT
ejpam-6813	118	14	a	a	NOUN
ejpam-6813	118	15	)	)	PUNCT
ejpam-6813	118	16	.	.	PUNCT
ejpam-6813	119	1	similarly	similarly	ADV
ejpam-6813	119	2	,	,	PUNCT
ejpam-6813	119	3	we	we	PRON
ejpam-6813	119	4	can	can	AUX
ejpam-6813	119	5	establish	establish	VERB
ejpam-6813	119	6	w	w	PROPN
ejpam-6813	119	7	(	(	PUNCT
ejpam-6813	119	8	1	1	NUM
ejpam-6813	119	9	)	)	PUNCT
ejpam-6813	119	10	n	n	PROPN
ejpam-6813	119	11	(	(	PUNCT
ejpam-6813	119	12	αa	αa	NOUN
ejpam-6813	119	13	)	)	PUNCT
ejpam-6813	119	14	=	=	SYM
ejpam-6813	120	1	w	w	PROPN
ejpam-6813	120	2	(	(	PUNCT
ejpam-6813	120	3	2	2	NUM
ejpam-6813	120	4	)	)	PUNCT
ejpam-6813	120	5	n	n	CCONJ
ejpam-6813	120	6	(	(	PUNCT
ejpam-6813	120	7	α̃a	α̃a	PROPN
ejpam-6813	120	8	)	)	PUNCT
ejpam-6813	121	1	=	=	SYM
ejpam-6813	121	2	w	w	PROPN
ejpam-6813	121	3	(	(	PUNCT
ejpam-6813	121	4	2	2	NUM
ejpam-6813	121	5	)	)	PUNCT
ejpam-6813	121	6	n	n	PROPN
ejpam-6813	121	7	(	(	PUNCT
ejpam-6813	121	8	αã	αã	NOUN
ejpam-6813	121	9	)	)	PUNCT
ejpam-6813	121	10	(	(	PUNCT
ejpam-6813	121	11	by	by	ADP
ejpam-6813	121	12	lemma	lemma	PROPN
ejpam-6813	121	13	1	1	NUM
ejpam-6813	121	14	)	)	PUNCT
ejpam-6813	121	15	=	=	SYM
ejpam-6813	121	16	|α|w(2	|α|w(2	NOUN
ejpam-6813	121	17	)	)	PUNCT
ejpam-6813	121	18	n	n	CCONJ
ejpam-6813	121	19	(	(	PUNCT
ejpam-6813	121	20	ã	ã	PROPN
ejpam-6813	121	21	)	)	PUNCT
ejpam-6813	121	22	=	=	PUNCT
ejpam-6813	121	23	|α|w(1	|α|w(1	X
ejpam-6813	121	24	)	)	PUNCT
ejpam-6813	121	25	n	n	CCONJ
ejpam-6813	121	26	(	(	PUNCT
ejpam-6813	121	27	a	a	NOUN
ejpam-6813	121	28	)	)	PUNCT
ejpam-6813	121	29	.	.	PUNCT
ejpam-6813	122	1	theorem	theorem	NOUN
ejpam-6813	122	2	2	2	NUM
ejpam-6813	122	3	.	.	X
ejpam-6813	122	4	for	for	ADP
ejpam-6813	122	5	a	a	PRON
ejpam-6813	122	6	=	=	PUNCT
ejpam-6813	123	1	[	[	X
ejpam-6813	123	2	ai	ai	NOUN
ejpam-6813	123	3	,	,	PUNCT
ejpam-6813	123	4	j	j	PROPN
ejpam-6813	123	5	]	]	PUNCT
ejpam-6813	123	6	∈	∈	PROPN
ejpam-6813	123	7	mm(mn	mm(mn	PROPN
ejpam-6813	123	8	)	)	PUNCT
ejpam-6813	123	9	,	,	PUNCT
ejpam-6813	123	10	if	if	SCONJ
ejpam-6813	123	11	either	either	PRON
ejpam-6813	123	12	w	w	PROPN
ejpam-6813	123	13	(	(	PUNCT
ejpam-6813	123	14	2	2	NUM
ejpam-6813	123	15	)	)	PUNCT
ejpam-6813	123	16	n	n	CCONJ
ejpam-6813	123	17	(	(	PUNCT
ejpam-6813	123	18	a	a	X
ejpam-6813	123	19	)	)	PUNCT
ejpam-6813	123	20	=	=	SYM
ejpam-6813	123	21	0	0	NUM
ejpam-6813	123	22	or	or	CCONJ
ejpam-6813	123	23	w	w	ADJ
ejpam-6813	123	24	(	(	PUNCT
ejpam-6813	123	25	1	1	NUM
ejpam-6813	123	26	)	)	PUNCT
ejpam-6813	123	27	n	n	CCONJ
ejpam-6813	123	28	(	(	PUNCT
ejpam-6813	123	29	a	a	X
ejpam-6813	123	30	)	)	PUNCT
ejpam-6813	123	31	=	=	SYM
ejpam-6813	123	32	0	0	NUM
ejpam-6813	123	33	,	,	PUNCT
ejpam-6813	123	34	then	then	ADV
ejpam-6813	123	35	necessarily	necessarily	ADV
ejpam-6813	123	36	a	a	PRON
ejpam-6813	123	37	=	=	SYM
ejpam-6813	123	38	0	0	NUM
ejpam-6813	123	39	.	.	PUNCT
ejpam-6813	124	1	proof	proof	NOUN
ejpam-6813	124	2	.	.	PUNCT
ejpam-6813	125	1	suppose	suppose	VERB
ejpam-6813	125	2	w	w	X
ejpam-6813	125	3	(	(	PUNCT
ejpam-6813	125	4	2	2	NUM
ejpam-6813	125	5	)	)	PUNCT
ejpam-6813	125	6	n	n	CCONJ
ejpam-6813	125	7	(	(	PUNCT
ejpam-6813	125	8	a	a	X
ejpam-6813	125	9	)	)	PUNCT
ejpam-6813	125	10	=	=	PUNCT
ejpam-6813	126	1	[	[	X
ejpam-6813	126	2	wn	wn	X
ejpam-6813	126	3	(	(	PUNCT
ejpam-6813	126	4	ai	ai	PROPN
ejpam-6813	126	5	,	,	PUNCT
ejpam-6813	126	6	j	j	PROPN
ejpam-6813	126	7	)	)	PUNCT
ejpam-6813	126	8	]	]	PUNCT
ejpam-6813	127	1	=	=	PUNCT
ejpam-6813	127	2	0	0	X
ejpam-6813	127	3	.	.	PUNCT
ejpam-6813	128	1	this	this	PRON
ejpam-6813	128	2	implies	imply	VERB
ejpam-6813	128	3	wn	wn	PROPN
ejpam-6813	128	4	(	(	PUNCT
ejpam-6813	128	5	ai	ai	PROPN
ejpam-6813	128	6	,	,	PUNCT
ejpam-6813	128	7	j	j	NOUN
ejpam-6813	128	8	)	)	PUNCT
ejpam-6813	128	9	=	=	SYM
ejpam-6813	128	10	0	0	NUM
ejpam-6813	129	1	for	for	ADP
ejpam-6813	129	2	all	all	DET
ejpam-6813	129	3	indices	index	NOUN
ejpam-6813	129	4	i	i	PRON
ejpam-6813	129	5	,	,	PUNCT
ejpam-6813	129	6	j	j	PROPN
ejpam-6813	129	7	=	=	SYM
ejpam-6813	129	8	1	1	NUM
ejpam-6813	129	9	,	,	PUNCT
ejpam-6813	129	10	2	2	NUM
ejpam-6813	129	11	,	,	PUNCT
ejpam-6813	129	12	.	.	PUNCT
ejpam-6813	129	13	.	.	PUNCT
ejpam-6813	129	14	.	.	PUNCT
ejpam-6813	130	1	,	,	PUNCT
ejpam-6813	130	2	m.	m.	NOUN
ejpam-6813	130	3	since	since	SCONJ
ejpam-6813	130	4	wn	wn	PROPN
ejpam-6813	130	5	(	(	PUNCT
ejpam-6813	130	6	a	a	PRON
ejpam-6813	130	7	)	)	PUNCT
ejpam-6813	130	8	functions	function	NOUN
ejpam-6813	130	9	as	as	ADP
ejpam-6813	130	10	a	a	DET
ejpam-6813	130	11	matrix	matrix	NOUN
ejpam-6813	130	12	norm	norm	NOUN
ejpam-6813	130	13	,	,	PUNCT
ejpam-6813	130	14	we	we	PRON
ejpam-6813	130	15	conclude	conclude	VERB
ejpam-6813	130	16	ai	ai	VERB
ejpam-6813	130	17	,	,	PUNCT
ejpam-6813	130	18	j	j	PROPN
ejpam-6813	130	19	=	=	PUNCT
ejpam-6813	130	20	0	0	PROPN
ejpam-6813	130	21	for	for	ADP
ejpam-6813	130	22	all	all	DET
ejpam-6813	130	23	i	i	PROPN
ejpam-6813	130	24	,	,	PUNCT
ejpam-6813	130	25	j	j	PROPN
ejpam-6813	130	26	=	=	SYM
ejpam-6813	130	27	1	1	NUM
ejpam-6813	130	28	,	,	PUNCT
ejpam-6813	130	29	2	2	NUM
ejpam-6813	130	30	,	,	PUNCT
ejpam-6813	130	31	.	.	PUNCT
ejpam-6813	130	32	.	.	PUNCT
ejpam-6813	131	1	.	.	PUNCT
ejpam-6813	132	1	,	,	PUNCT
ejpam-6813	132	2	m	m	AUX
ejpam-6813	132	3	,	,	PUNCT
ejpam-6813	132	4	yielding	yield	VERB
ejpam-6813	132	5	a	a	DET
ejpam-6813	132	6	=	=	NOUN
ejpam-6813	132	7	0	0	NUM
ejpam-6813	132	8	.	.	PUNCT
ejpam-6813	133	1	similarly	similarly	ADV
ejpam-6813	133	2	,	,	PUNCT
ejpam-6813	133	3	if	if	SCONJ
ejpam-6813	133	4	w	w	PROPN
ejpam-6813	133	5	(	(	PUNCT
ejpam-6813	133	6	1	1	NUM
ejpam-6813	133	7	)	)	PUNCT
ejpam-6813	133	8	n	n	CCONJ
ejpam-6813	133	9	(	(	PUNCT
ejpam-6813	133	10	a	a	X
ejpam-6813	133	11	)	)	PUNCT
ejpam-6813	133	12	=	=	PUNCT
ejpam-6813	134	1	[	[	X
ejpam-6813	134	2	wn	wn	X
ejpam-6813	134	3	(	(	PUNCT
ejpam-6813	134	4	gl	gl	PROPN
ejpam-6813	134	5	,	,	PUNCT
ejpam-6813	134	6	k	k	NOUN
ejpam-6813	134	7	)	)	PUNCT
ejpam-6813	134	8	]	]	PUNCT
ejpam-6813	135	1	=	=	PUNCT
ejpam-6813	135	2	0	0	NUM
ejpam-6813	135	3	,	,	PUNCT
ejpam-6813	135	4	then	then	ADV
ejpam-6813	135	5	wn	wn	PROPN
ejpam-6813	135	6	(	(	PUNCT
ejpam-6813	135	7	gl	gl	PROPN
ejpam-6813	135	8	,	,	PUNCT
ejpam-6813	135	9	k	k	NOUN
ejpam-6813	135	10	)	)	PUNCT
ejpam-6813	135	11	=	=	SYM
ejpam-6813	135	12	0	0	NUM
ejpam-6813	135	13	for	for	ADP
ejpam-6813	135	14	all	all	DET
ejpam-6813	135	15	indices	index	NOUN
ejpam-6813	135	16	l	l	NOUN
ejpam-6813	135	17	,	,	PUNCT
ejpam-6813	135	18	k	k	PROPN
ejpam-6813	135	19	=	=	SYM
ejpam-6813	135	20	1	1	NUM
ejpam-6813	135	21	,	,	PUNCT
ejpam-6813	135	22	2	2	NUM
ejpam-6813	135	23	,	,	PUNCT
ejpam-6813	135	24	.	.	PUNCT
ejpam-6813	135	25	.	.	PUNCT
ejpam-6813	135	26	.	.	PUNCT
ejpam-6813	136	1	,	,	PUNCT
ejpam-6813	136	2	n.	n.	AUX
ejpam-6813	136	3	again	again	ADV
ejpam-6813	136	4	using	use	VERB
ejpam-6813	136	5	the	the	DET
ejpam-6813	136	6	norm	norm	NOUN
ejpam-6813	136	7	property	property	NOUN
ejpam-6813	136	8	of	of	ADP
ejpam-6813	136	9	wn	wn	PROPN
ejpam-6813	136	10	(	(	PUNCT
ejpam-6813	136	11	a	a	NOUN
ejpam-6813	136	12	)	)	PUNCT
ejpam-6813	136	13	,	,	PUNCT
ejpam-6813	136	14	we	we	PRON
ejpam-6813	136	15	get	get	VERB
ejpam-6813	136	16	gl	gl	PROPN
ejpam-6813	136	17	,	,	PUNCT
ejpam-6813	136	18	k	k	PROPN
ejpam-6813	137	1	=	=	PUNCT
ejpam-6813	137	2	0	0	NUM
ejpam-6813	137	3	for	for	ADP
ejpam-6813	137	4	all	all	DET
ejpam-6813	137	5	l	l	NOUN
ejpam-6813	137	6	,	,	PUNCT
ejpam-6813	137	7	k	k	PROPN
ejpam-6813	137	8	=	=	SYM
ejpam-6813	137	9	1	1	NUM
ejpam-6813	137	10	,	,	PUNCT
ejpam-6813	137	11	2	2	NUM
ejpam-6813	137	12	,	,	PUNCT
ejpam-6813	137	13	.	.	PUNCT
ejpam-6813	137	14	.	.	PUNCT
ejpam-6813	138	1	.	.	PUNCT
ejpam-6813	139	1	,	,	PUNCT
ejpam-6813	139	2	n	n	CCONJ
ejpam-6813	139	3	,	,	PUNCT
ejpam-6813	139	4	which	which	PRON
ejpam-6813	139	5	means	mean	VERB
ejpam-6813	139	6	ã	ã	PROPN
ejpam-6813	139	7	=	=	SYM
ejpam-6813	139	8	0	0	PROPN
ejpam-6813	139	9	.	.	PUNCT
ejpam-6813	140	1	since	since	SCONJ
ejpam-6813	140	2	a	a	PRON
ejpam-6813	140	3	and	and	CCONJ
ejpam-6813	140	4	ã	ã	PROPN
ejpam-6813	140	5	are	be	AUX
ejpam-6813	140	6	unitarily	unitarily	ADV
ejpam-6813	140	7	similar	similar	ADJ
ejpam-6813	140	8	,	,	PUNCT
ejpam-6813	140	9	we	we	PRON
ejpam-6813	140	10	conclude	conclude	VERB
ejpam-6813	140	11	a	a	DET
ejpam-6813	140	12	=	=	NOUN
ejpam-6813	140	13	0	0	X
ejpam-6813	140	14	.	.	PUNCT
ejpam-6813	141	1	we	we	PRON
ejpam-6813	141	2	introduce	introduce	VERB
ejpam-6813	141	3	the	the	DET
ejpam-6813	141	4	concept	concept	NOUN
ejpam-6813	141	5	of	of	ADP
ejpam-6813	141	6	partial	partial	ADJ
ejpam-6813	141	7	normality	normality	NOUN
ejpam-6813	141	8	for	for	ADP
ejpam-6813	141	9	block	block	NOUN
ejpam-6813	141	10	matrices	matrix	NOUN
ejpam-6813	141	11	,	,	PUNCT
ejpam-6813	141	12	which	which	PRON
ejpam-6813	141	13	will	will	AUX
ejpam-6813	141	14	prove	prove	VERB
ejpam-6813	141	15	crucial	crucial	ADJ
ejpam-6813	141	16	for	for	ADP
ejpam-6813	141	17	establishing	establish	VERB
ejpam-6813	141	18	several	several	ADJ
ejpam-6813	141	19	key	key	ADJ
ejpam-6813	141	20	results	result	NOUN
ejpam-6813	141	21	.	.	PUNCT
ejpam-6813	142	1	definition	definition	NOUN
ejpam-6813	142	2	3	3	NUM
ejpam-6813	142	3	.	.	PUNCT
ejpam-6813	143	1	a	a	DET
ejpam-6813	143	2	matrix	matrix	NOUN
ejpam-6813	143	3	a	a	PRON
ejpam-6813	144	1	=	=	PUNCT
ejpam-6813	145	1	[	[	X
ejpam-6813	145	2	ai	ai	NOUN
ejpam-6813	145	3	,	,	PUNCT
ejpam-6813	145	4	j	j	PROPN
ejpam-6813	145	5	]	]	PUNCT
ejpam-6813	145	6	∈	∈	PROPN
ejpam-6813	145	7	mm(mn	mm(mn	PROPN
ejpam-6813	145	8	)	)	PUNCT
ejpam-6813	145	9	is	be	AUX
ejpam-6813	145	10	called	call	VERB
ejpam-6813	145	11	partially	partially	ADV
ejpam-6813	145	12	normal	normal	ADJ
ejpam-6813	145	13	when	when	SCONJ
ejpam-6813	145	14	each	each	DET
ejpam-6813	145	15	block	block	NOUN
ejpam-6813	145	16	ai	ai	VERB
ejpam-6813	145	17	,	,	PUNCT
ejpam-6813	145	18	j	j	PROPN
ejpam-6813	145	19	is	be	AUX
ejpam-6813	145	20	normal	normal	ADJ
ejpam-6813	145	21	for	for	ADP
ejpam-6813	145	22	all	all	DET
ejpam-6813	145	23	1	1	NUM
ejpam-6813	145	24	≤	≤	NUM
ejpam-6813	145	25	i	i	PRON
ejpam-6813	145	26	,	,	PUNCT
ejpam-6813	145	27	j	j	PROPN
ejpam-6813	145	28	≤	≤	PROPN
ejpam-6813	145	29	m.	m.	NOUN
ejpam-6813	145	30	the	the	DET
ejpam-6813	145	31	concept	concept	NOUN
ejpam-6813	145	32	of	of	ADP
ejpam-6813	145	33	partial	partial	ADJ
ejpam-6813	145	34	normality	normality	NOUN
ejpam-6813	145	35	provides	provide	VERB
ejpam-6813	145	36	a	a	DET
ejpam-6813	145	37	natural	natural	ADJ
ejpam-6813	145	38	generalization	generalization	NOUN
ejpam-6813	145	39	of	of	ADP
ejpam-6813	145	40	matrix	matrix	NOUN
ejpam-6813	145	41	normality	normality	NOUN
ejpam-6813	145	42	to	to	ADP
ejpam-6813	145	43	the	the	DET
ejpam-6813	145	44	block	block	NOUN
ejpam-6813	145	45	setting	setting	NOUN
ejpam-6813	145	46	.	.	PUNCT
ejpam-6813	146	1	this	this	DET
ejpam-6813	146	2	property	property	NOUN
ejpam-6813	146	3	allows	allow	VERB
ejpam-6813	146	4	us	we	PRON
ejpam-6813	146	5	to	to	PART
ejpam-6813	146	6	extend	extend	VERB
ejpam-6813	146	7	many	many	ADJ
ejpam-6813	146	8	classical	classical	ADJ
ejpam-6813	146	9	results	result	NOUN
ejpam-6813	146	10	about	about	ADP
ejpam-6813	146	11	normal	normal	ADJ
ejpam-6813	146	12	matrices	matrix	NOUN
ejpam-6813	146	13	to	to	ADP
ejpam-6813	146	14	the	the	DET
ejpam-6813	146	15	block	block	NOUN
ejpam-6813	146	16	matrix	matrix	NOUN
ejpam-6813	146	17	framework	framework	NOUN
ejpam-6813	146	18	while	while	SCONJ
ejpam-6813	146	19	maintaining	maintain	VERB
ejpam-6813	146	20	their	their	PRON
ejpam-6813	146	21	essential	essential	ADJ
ejpam-6813	146	22	characteristics	characteristic	NOUN
ejpam-6813	146	23	.	.	PUNCT
ejpam-6813	147	1	theorem	theorem	NOUN
ejpam-6813	147	2	3	3	NUM
ejpam-6813	147	3	.	.	X
ejpam-6813	147	4	for	for	ADP
ejpam-6813	147	5	a	a	DET
ejpam-6813	147	6	partially	partially	ADV
ejpam-6813	147	7	normal	normal	ADJ
ejpam-6813	147	8	matrix	matrix	NOUN
ejpam-6813	148	1	a	a	PRON
ejpam-6813	148	2	=	=	PUNCT
ejpam-6813	149	1	[	[	X
ejpam-6813	149	2	ai	ai	NOUN
ejpam-6813	149	3	,	,	PUNCT
ejpam-6813	149	4	j	j	PROPN
ejpam-6813	149	5	]	]	PUNCT
ejpam-6813	149	6	∈	∈	PROPN
ejpam-6813	149	7	mm(mn	mm(mn	PROPN
ejpam-6813	149	8	)	)	PUNCT
ejpam-6813	149	9	,	,	PUNCT
ejpam-6813	149	10	we	we	PRON
ejpam-6813	149	11	have	have	VERB
ejpam-6813	149	12	w(2)(a	w(2)(a	NOUN
ejpam-6813	149	13	◦	◦	NOUN
ejpam-6813	149	14	h	h	NOUN
ejpam-6813	149	15	)	)	PUNCT
ejpam-6813	149	16	=	=	SYM
ejpam-6813	150	1	(	(	PUNCT
ejpam-6813	150	2	w(2)(a))	w(2)(a))	VERB
ejpam-6813	150	3	◦	◦	NOUN
ejpam-6813	150	4	h.	h.	NOUN
ejpam-6813	150	5	proof	proof	NOUN
ejpam-6813	150	6	.	.	PUNCT
ejpam-6813	151	1	given	give	VERB
ejpam-6813	151	2	a	a	PRON
ejpam-6813	151	3	=	=	PRON
ejpam-6813	152	1	[	[	X
ejpam-6813	152	2	ai	ai	NOUN
ejpam-6813	152	3	,	,	PUNCT
ejpam-6813	152	4	j	j	PROPN
ejpam-6813	152	5	]	]	PUNCT
ejpam-6813	152	6	∈	∈	PROPN
ejpam-6813	152	7	mm(mn	mm(mn	PROPN
ejpam-6813	152	8	)	)	PUNCT
ejpam-6813	152	9	where	where	SCONJ
ejpam-6813	152	10	each	each	PRON
ejpam-6813	152	11	ai	ai	VERB
ejpam-6813	152	12	,	,	PUNCT
ejpam-6813	152	13	j	j	PROPN
ejpam-6813	152	14	is	be	AUX
ejpam-6813	152	15	normal	normal	ADJ
ejpam-6813	152	16	,	,	PUNCT
ejpam-6813	152	17	we	we	PRON
ejpam-6813	152	18	use	use	VERB
ejpam-6813	152	19	the	the	DET
ejpam-6813	152	20	property	property	NOUN
ejpam-6813	152	21	that	that	PRON
ejpam-6813	152	22	for	for	ADP
ejpam-6813	152	23	normal	normal	ADJ
ejpam-6813	152	24	matrices	matrix	NOUN
ejpam-6813	152	25	w(bh	w(bh	NOUN
ejpam-6813	152	26	)	)	PUNCT
ejpam-6813	152	27	=	=	PUNCT
ejpam-6813	152	28	wh(b	wh(b	X
ejpam-6813	152	29	)	)	PUNCT
ejpam-6813	152	30	for	for	ADP
ejpam-6813	152	31	h	h	NOUN
ejpam-6813	152	32	>	>	X
ejpam-6813	152	33	0	0	PROPN
ejpam-6813	152	34	.	.	PUNCT
ejpam-6813	153	1	therefore	therefore	ADV
ejpam-6813	153	2	:	:	PUNCT
ejpam-6813	153	3	w(2)(a	w(2)(a	PROPN
ejpam-6813	153	4	◦	◦	NOUN
ejpam-6813	153	5	h	h	NOUN
ejpam-6813	153	6	)	)	PUNCT
ejpam-6813	153	7	=	=	NOUN
ejpam-6813	154	1	[	[	X
ejpam-6813	154	2	w(ah	w(ah	NOUN
ejpam-6813	154	3	i	i	PRON
ejpam-6813	154	4	,	,	PUNCT
ejpam-6813	154	5	j	j	PROPN
ejpam-6813	154	6	)	)	PUNCT
ejpam-6813	154	7	]	]	PUNCT
ejpam-6813	155	1	m	m	VERB
ejpam-6813	155	2	i	i	PRON
ejpam-6813	155	3	,	,	PUNCT
ejpam-6813	155	4	j=1	j=1	PROPN
ejpam-6813	156	1	=	=	PUNCT
ejpam-6813	157	1	[	[	X
ejpam-6813	157	2	wh(ai	wh(ai	PROPN
ejpam-6813	157	3	,	,	PUNCT
ejpam-6813	157	4	j	j	NOUN
ejpam-6813	157	5	)	)	PUNCT
ejpam-6813	157	6	]	]	PUNCT
ejpam-6813	158	1	m	m	VERB
ejpam-6813	158	2	i	i	PRON
ejpam-6813	158	3	,	,	PUNCT
ejpam-6813	158	4	j=1	j=1	NOUN
ejpam-6813	158	5	=	=	PUNCT
ejpam-6813	159	1	(	(	PUNCT
ejpam-6813	159	2	[	[	X
ejpam-6813	159	3	w(ai	w(ai	PROPN
ejpam-6813	159	4	,	,	PUNCT
ejpam-6813	159	5	j	j	PROPN
ejpam-6813	159	6	)	)	PUNCT
ejpam-6813	159	7	]	]	PUNCT
ejpam-6813	160	1	m	m	VERB
ejpam-6813	160	2	i	i	PRON
ejpam-6813	160	3	,	,	PUNCT
ejpam-6813	160	4	j=1	j=1	NOUN
ejpam-6813	160	5	)	)	PUNCT
ejpam-6813	160	6	◦	◦	NOUN
ejpam-6813	160	7	h	h	NOUN
ejpam-6813	160	8	=	=	SYM
ejpam-6813	160	9	(	(	PUNCT
ejpam-6813	160	10	w(2)(a))	w(2)(a))	VERB
ejpam-6813	160	11	◦	◦	NOUN
ejpam-6813	160	12	h.	h.	NOUN
ejpam-6813	160	13	this	this	DET
ejpam-6813	160	14	result	result	NOUN
ejpam-6813	160	15	extends	extend	VERB
ejpam-6813	160	16	the	the	DET
ejpam-6813	160	17	well	well	ADV
ejpam-6813	160	18	-	-	PUNCT
ejpam-6813	160	19	known	know	VERB
ejpam-6813	160	20	power	power	NOUN
ejpam-6813	160	21	inequality	inequality	NOUN
ejpam-6813	160	22	w(ak	w(ak	PROPN
ejpam-6813	160	23	)	)	PUNCT
ejpam-6813	160	24	=	=	SYM
ejpam-6813	160	25	(	(	PUNCT
ejpam-6813	160	26	w(a))k	w(a))k	ADP
ejpam-6813	160	27	for	for	ADP
ejpam-6813	160	28	normal	normal	ADJ
ejpam-6813	160	29	matrices	matrix	NOUN
ejpam-6813	160	30	a	a	DET
ejpam-6813	160	31	∈	∈	NOUN
ejpam-6813	160	32	mm(c	mm(c	NOUN
ejpam-6813	160	33	)	)	PUNCT
ejpam-6813	160	34	and	and	CCONJ
ejpam-6813	160	35	k	k	PROPN
ejpam-6813	160	36	∈	∈	PROPN
ejpam-6813	160	37	(	(	PUNCT
ejpam-6813	160	38	0,∞	0,∞	NOUN
ejpam-6813	160	39	)	)	PUNCT
ejpam-6813	160	40	to	to	ADP
ejpam-6813	160	41	the	the	DET
ejpam-6813	160	42	block	block	NOUN
ejpam-6813	160	43	matrix	matrix	NOUN
ejpam-6813	160	44	context	context	NOUN
ejpam-6813	160	45	using	use	VERB
ejpam-6813	160	46	hadamard	hadamard	ADJ
ejpam-6813	160	47	operations	operation	NOUN
ejpam-6813	160	48	.	.	PUNCT
ejpam-6813	161	1	r.	r.	PROPN
ejpam-6813	161	2	al	al	PROPN
ejpam-6813	161	3	-	-	PROPN
ejpam-6813	161	4	naimi	naimi	PROPN
ejpam-6813	161	5	et	et	PROPN
ejpam-6813	161	6	al	al	PROPN
ejpam-6813	161	7	.	.	PUNCT
ejpam-6813	161	8	/	/	SYM
ejpam-6813	161	9	eur	eur	PROPN
ejpam-6813	161	10	.	.	PUNCT
ejpam-6813	162	1	j.	j.	PROPN
ejpam-6813	162	2	pure	pure	PROPN
ejpam-6813	162	3	appl	appl	PROPN
ejpam-6813	162	4	.	.	PROPN
ejpam-6813	162	5	math	math	PROPN
ejpam-6813	162	6	,	,	PUNCT
ejpam-6813	162	7	18	18	NUM
ejpam-6813	162	8	(	(	PUNCT
ejpam-6813	162	9	4	4	NUM
ejpam-6813	162	10	)	)	PUNCT
ejpam-6813	162	11	(	(	PUNCT
ejpam-6813	162	12	2025	2025	NUM
ejpam-6813	162	13	)	)	PUNCT
ejpam-6813	162	14	,	,	PUNCT
ejpam-6813	162	15	6813	6813	NUM
ejpam-6813	162	16	6	6	NUM
ejpam-6813	162	17	of	of	ADP
ejpam-6813	162	18	15	15	NUM
ejpam-6813	162	19	3	3	NUM
ejpam-6813	162	20	.	.	PUNCT
ejpam-6813	162	21	unitary	unitary	ADJ
ejpam-6813	162	22	invariance	invariance	NOUN
ejpam-6813	162	23	and	and	CCONJ
ejpam-6813	162	24	structural	structural	ADJ
ejpam-6813	162	25	properties	property	NOUN
ejpam-6813	162	26	the	the	DET
ejpam-6813	162	27	behavior	behavior	NOUN
ejpam-6813	162	28	of	of	ADP
ejpam-6813	162	29	partial	partial	ADJ
ejpam-6813	162	30	numerical	numerical	ADJ
ejpam-6813	162	31	radius	radius	NOUN
ejpam-6813	162	32	under	under	ADP
ejpam-6813	162	33	various	various	ADJ
ejpam-6813	162	34	transformations	transformation	NOUN
ejpam-6813	162	35	provides	provide	VERB
ejpam-6813	162	36	important	important	ADJ
ejpam-6813	162	37	insights	insight	NOUN
ejpam-6813	162	38	into	into	ADP
ejpam-6813	162	39	the	the	DET
ejpam-6813	162	40	geometric	geometric	ADJ
ejpam-6813	162	41	structure	structure	NOUN
ejpam-6813	162	42	of	of	ADP
ejpam-6813	162	43	block	block	NOUN
ejpam-6813	162	44	matrices	matrix	NOUN
ejpam-6813	162	45	.	.	PUNCT
ejpam-6813	163	1	theorem	theorem	ADJ
ejpam-6813	163	2	4	4	NUM
ejpam-6813	163	3	.	.	PUNCT
ejpam-6813	163	4	consider	consider	VERB
ejpam-6813	163	5	matrices	matrix	NOUN
ejpam-6813	164	1	a	a	PRON
ejpam-6813	164	2	=	=	NOUN
ejpam-6813	165	1	[	[	X
ejpam-6813	165	2	ai	ai	NOUN
ejpam-6813	165	3	,	,	PUNCT
ejpam-6813	165	4	j	j	PROPN
ejpam-6813	165	5	]	]	PUNCT
ejpam-6813	165	6	∈	∈	PROPN
ejpam-6813	165	7	mm(mn	mm(mn	PROPN
ejpam-6813	165	8	)	)	PUNCT
ejpam-6813	165	9	,	,	PUNCT
ejpam-6813	165	10	u	u	NOUN
ejpam-6813	166	1	=	=	PUNCT
ejpam-6813	167	1	[	[	X
ejpam-6813	167	2	ui	ui	PROPN
ejpam-6813	167	3	,	,	PUNCT
ejpam-6813	167	4	j	j	PROPN
ejpam-6813	167	5	]	]	PUNCT
ejpam-6813	167	6	∈	∈	PROPN
ejpam-6813	167	7	mm(mn	mm(mn	PROPN
ejpam-6813	167	8	)	)	PUNCT
ejpam-6813	167	9	,	,	PUNCT
ejpam-6813	167	10	and	and	CCONJ
ejpam-6813	167	11	v	v	X
ejpam-6813	167	12	=	=	SYM
ejpam-6813	167	13	[	[	X
ejpam-6813	167	14	vi	vi	PROPN
ejpam-6813	167	15	,	,	PUNCT
ejpam-6813	167	16	j	j	NOUN
ejpam-6813	167	17	]	]	PUNCT
ejpam-6813	167	18	∈	∈	PROPN
ejpam-6813	167	19	mm(mn	mm(mn	PROPN
ejpam-6813	167	20	)	)	PUNCT
ejpam-6813	167	21	where	where	SCONJ
ejpam-6813	167	22	each	each	DET
ejpam-6813	167	23	ui	ui	PROPN
ejpam-6813	167	24	,	,	PUNCT
ejpam-6813	167	25	j	j	PROPN
ejpam-6813	167	26	and	and	CCONJ
ejpam-6813	167	27	vi	vi	PROPN
ejpam-6813	167	28	,	,	PUNCT
ejpam-6813	167	29	j	j	PROPN
ejpam-6813	167	30	is	be	AUX
ejpam-6813	167	31	unitary	unitary	ADJ
ejpam-6813	167	32	for	for	ADP
ejpam-6813	167	33	1	1	NUM
ejpam-6813	167	34	≤	≤	NUM
ejpam-6813	168	1	i	i	PRON
ejpam-6813	168	2	,	,	PUNCT
ejpam-6813	168	3	j	j	PROPN
ejpam-6813	168	4	≤	≤	PROPN
ejpam-6813	168	5	m.	m.	NOUN
ejpam-6813	168	6	then	then	ADV
ejpam-6813	168	7	w	w	PROPN
ejpam-6813	168	8	(	(	PUNCT
ejpam-6813	168	9	2	2	NUM
ejpam-6813	168	10	)	)	PUNCT
ejpam-6813	168	11	n	n	CCONJ
ejpam-6813	168	12	(	(	PUNCT
ejpam-6813	168	13	a	a	X
ejpam-6813	168	14	)	)	PUNCT
ejpam-6813	168	15	=	=	SYM
ejpam-6813	168	16	w	w	PROPN
ejpam-6813	168	17	(	(	PUNCT
ejpam-6813	168	18	2	2	NUM
ejpam-6813	168	19	)	)	PUNCT
ejpam-6813	168	20	n	n	NOUN
ejpam-6813	168	21	(	(	PUNCT
ejpam-6813	168	22	u	u	NOUN
ejpam-6813	168	23	◦	◦	VERB
ejpam-6813	168	24	a	a	DET
ejpam-6813	168	25	◦	◦	NOUN
ejpam-6813	168	26	v	v	NOUN
ejpam-6813	168	27	)	)	PUNCT
ejpam-6813	168	28	when	when	SCONJ
ejpam-6813	168	29	n	n	CCONJ
ejpam-6813	168	30	(	(	PUNCT
ejpam-6813	168	31	·	·	PUNCT
ejpam-6813	168	32	)	)	PUNCT
ejpam-6813	168	33	is	be	AUX
ejpam-6813	168	34	a	a	DET
ejpam-6813	168	35	unitarily	unitarily	ADV
ejpam-6813	168	36	invariant	invariant	ADJ
ejpam-6813	168	37	norm	norm	NOUN
ejpam-6813	168	38	.	.	PUNCT
ejpam-6813	169	1	proof	proof	NOUN
ejpam-6813	169	2	.	.	PUNCT
ejpam-6813	170	1	by	by	ADP
ejpam-6813	170	2	the	the	DET
ejpam-6813	170	3	unitary	unitary	ADJ
ejpam-6813	170	4	invariance	invariance	NOUN
ejpam-6813	170	5	of	of	ADP
ejpam-6813	170	6	the	the	DET
ejpam-6813	170	7	generalized	generalized	ADJ
ejpam-6813	170	8	numerical	numerical	ADJ
ejpam-6813	170	9	radius	radius	NOUN
ejpam-6813	170	10	:	:	PUNCT
ejpam-6813	170	11	w	w	X
ejpam-6813	170	12	(	(	PUNCT
ejpam-6813	170	13	2	2	NUM
ejpam-6813	170	14	)	)	PUNCT
ejpam-6813	170	15	n	n	NOUN
ejpam-6813	170	16	(	(	PUNCT
ejpam-6813	170	17	u	u	NOUN
ejpam-6813	170	18	◦	◦	VERB
ejpam-6813	170	19	a	a	DET
ejpam-6813	170	20	◦	◦	NOUN
ejpam-6813	170	21	v	v	NOUN
ejpam-6813	170	22	)	)	PUNCT
ejpam-6813	170	23	=	=	PUNCT
ejpam-6813	171	1	[	[	X
ejpam-6813	171	2	wn	wn	X
ejpam-6813	171	3	(	(	PUNCT
ejpam-6813	171	4	ui	ui	PROPN
ejpam-6813	171	5	,	,	PUNCT
ejpam-6813	171	6	jai	jai	PROPN
ejpam-6813	171	7	,	,	PUNCT
ejpam-6813	171	8	jvi	jvi	ADV
ejpam-6813	171	9	,	,	PUNCT
ejpam-6813	171	10	j	j	PROPN
ejpam-6813	171	11	)	)	PUNCT
ejpam-6813	171	12	]	]	PUNCT
ejpam-6813	171	13	m	m	VERB
ejpam-6813	171	14	i	i	PRON
ejpam-6813	171	15	,	,	PUNCT
ejpam-6813	171	16	j=1	j=1	PROPN
ejpam-6813	171	17	=	=	PUNCT
ejpam-6813	172	1	[	[	X
ejpam-6813	172	2	wn	wn	X
ejpam-6813	172	3	(	(	PUNCT
ejpam-6813	172	4	ai	ai	PROPN
ejpam-6813	172	5	,	,	PUNCT
ejpam-6813	172	6	j	j	PROPN
ejpam-6813	172	7	)	)	PUNCT
ejpam-6813	172	8	]	]	PUNCT
ejpam-6813	173	1	m	m	VERB
ejpam-6813	173	2	i	i	PRON
ejpam-6813	173	3	,	,	PUNCT
ejpam-6813	173	4	j=1	j=1	PROPN
ejpam-6813	173	5	=	=	PUNCT
ejpam-6813	173	6	w	w	PROPN
ejpam-6813	173	7	(	(	PUNCT
ejpam-6813	173	8	2	2	NUM
ejpam-6813	173	9	)	)	PUNCT
ejpam-6813	173	10	n	n	CCONJ
ejpam-6813	173	11	(	(	PUNCT
ejpam-6813	173	12	a	a	NOUN
ejpam-6813	173	13	)	)	PUNCT
ejpam-6813	173	14	.	.	PUNCT
ejpam-6813	174	1	this	this	DET
ejpam-6813	174	2	theorem	theorem	NOUN
ejpam-6813	174	3	establishes	establish	VERB
ejpam-6813	174	4	that	that	SCONJ
ejpam-6813	174	5	our	our	PRON
ejpam-6813	174	6	partial	partial	ADJ
ejpam-6813	174	7	numerical	numerical	ADJ
ejpam-6813	174	8	radius	radius	PROPN
ejpam-6813	174	9	constructions	construction	NOUN
ejpam-6813	174	10	respect	respect	VERB
ejpam-6813	174	11	the	the	DET
ejpam-6813	174	12	fundamental	fundamental	ADJ
ejpam-6813	174	13	unitary	unitary	ADJ
ejpam-6813	174	14	invariance	invariance	NOUN
ejpam-6813	174	15	property	property	NOUN
ejpam-6813	174	16	that	that	PRON
ejpam-6813	174	17	is	be	AUX
ejpam-6813	174	18	central	central	ADJ
ejpam-6813	174	19	to	to	ADP
ejpam-6813	174	20	numerical	numerical	ADJ
ejpam-6813	174	21	radius	radius	PROPN
ejpam-6813	174	22	theory	theory	NOUN
ejpam-6813	174	23	.	.	PUNCT
ejpam-6813	175	1	corollary	corollary	ADJ
ejpam-6813	175	2	1	1	NUM
ejpam-6813	175	3	.	.	PUNCT
ejpam-6813	176	1	under	under	ADP
ejpam-6813	176	2	the	the	DET
ejpam-6813	176	3	conditions	condition	NOUN
ejpam-6813	176	4	of	of	ADP
ejpam-6813	176	5	theorem	theorem	ADJ
ejpam-6813	176	6	4	4	NUM
ejpam-6813	176	7	,	,	PUNCT
ejpam-6813	176	8	setting	set	VERB
ejpam-6813	176	9	n	n	CCONJ
ejpam-6813	176	10	(	(	PUNCT
ejpam-6813	176	11	·	·	PUNCT
ejpam-6813	176	12	)	)	PUNCT
ejpam-6813	176	13	=	=	PUNCT
ejpam-6813	176	14	‖	‖	PROPN
ejpam-6813	176	15	·	·	PUNCT
ejpam-6813	176	16	‖	‖	ADJ
ejpam-6813	176	17	yields	yield	NOUN
ejpam-6813	176	18	w(2)(a	w(2)(a	NOUN
ejpam-6813	176	19	)	)	PUNCT
ejpam-6813	176	20	=	=	VERB
ejpam-6813	176	21	w(2)(u	w(2)(u	ADJ
ejpam-6813	176	22	◦	◦	NOUN
ejpam-6813	176	23	a	a	DET
ejpam-6813	176	24	◦	◦	NOUN
ejpam-6813	176	25	v	v	NUM
ejpam-6813	176	26	)	)	PUNCT
ejpam-6813	176	27	.	.	PUNCT
ejpam-6813	177	1	block	block	NOUN
ejpam-6813	177	2	unitary	unitary	ADJ
ejpam-6813	177	3	transformations	transformation	NOUN
ejpam-6813	177	4	provide	provide	VERB
ejpam-6813	177	5	a	a	DET
ejpam-6813	177	6	natural	natural	ADJ
ejpam-6813	177	7	framework	framework	NOUN
ejpam-6813	177	8	for	for	ADP
ejpam-6813	177	9	analyzing	analyze	VERB
ejpam-6813	177	10	the	the	DET
ejpam-6813	177	11	structure	structure	NOUN
ejpam-6813	177	12	of	of	ADP
ejpam-6813	177	13	partial	partial	ADJ
ejpam-6813	177	14	numerical	numerical	ADJ
ejpam-6813	177	15	radius	radius	NOUN
ejpam-6813	177	16	.	.	PUNCT
ejpam-6813	178	1	theorem	theorem	NOUN
ejpam-6813	178	2	5	5	NUM
ejpam-6813	178	3	.	.	PUNCT
ejpam-6813	179	1	let	let	VERB
ejpam-6813	179	2	a	a	DET
ejpam-6813	179	3	=	=	PUNCT
ejpam-6813	179	4	[	[	X
ejpam-6813	179	5	aij	aij	X
ejpam-6813	179	6	]	]	PUNCT
ejpam-6813	179	7	∈	∈	PROPN
ejpam-6813	179	8	mm(mn	mm(mn	PROPN
ejpam-6813	179	9	)	)	PUNCT
ejpam-6813	179	10	and	and	CCONJ
ejpam-6813	179	11	let	let	VERB
ejpam-6813	179	12	p	p	NOUN
ejpam-6813	179	13	=	=	VERB
ejpam-6813	179	14	diag(u1,1	diag(u1,1	PROPN
ejpam-6813	179	15	,	,	PUNCT
ejpam-6813	179	16	u2,2	u2,2	PROPN
ejpam-6813	179	17	,	,	PUNCT
ejpam-6813	179	18	.	.	PUNCT
ejpam-6813	179	19	.	.	PUNCT
ejpam-6813	180	1	.	.	PUNCT
ejpam-6813	181	1	,	,	PUNCT
ejpam-6813	181	2	um	um	INTJ
ejpam-6813	181	3	,	,	PUNCT
ejpam-6813	181	4	m	m	VERB
ejpam-6813	181	5	)	)	PUNCT
ejpam-6813	181	6	be	be	VERB
ejpam-6813	181	7	a	a	DET
ejpam-6813	181	8	block	block	NOUN
ejpam-6813	181	9	diagonal	diagonal	ADJ
ejpam-6813	181	10	unitary	unitary	ADJ
ejpam-6813	181	11	matrix	matrix	NOUN
ejpam-6813	181	12	where	where	SCONJ
ejpam-6813	181	13	uk	uk	PROPN
ejpam-6813	181	14	,	,	PUNCT
ejpam-6813	181	15	k	k	PROPN
ejpam-6813	181	16	∈	∈	PROPN
ejpam-6813	181	17	mn(c	mn(c	X
ejpam-6813	181	18	)	)	PUNCT
ejpam-6813	181	19	.	.	PUNCT
ejpam-6813	182	1	then	then	ADV
ejpam-6813	182	2	:	:	PUNCT
ejpam-6813	182	3	w	w	X
ejpam-6813	182	4	(	(	PUNCT
ejpam-6813	182	5	2	2	NUM
ejpam-6813	182	6	)	)	PUNCT
ejpam-6813	182	7	n	n	CCONJ
ejpam-6813	182	8	(	(	PUNCT
ejpam-6813	182	9	pap	pap	NOUN
ejpam-6813	182	10	∗	∗	NOUN
ejpam-6813	182	11	)	)	PUNCT
ejpam-6813	183	1	=	=	SYM
ejpam-6813	183	2	w	w	NOUN
ejpam-6813	183	3	(	(	PUNCT
ejpam-6813	183	4	2	2	NUM
ejpam-6813	183	5	)	)	PUNCT
ejpam-6813	183	6	n	n	CCONJ
ejpam-6813	183	7	(	(	PUNCT
ejpam-6813	183	8	a	a	NOUN
ejpam-6813	183	9	)	)	PUNCT
ejpam-6813	183	10	,	,	PUNCT
ejpam-6813	183	11	where	where	SCONJ
ejpam-6813	183	12	n	n	CCONJ
ejpam-6813	183	13	(	(	PUNCT
ejpam-6813	183	14	·	·	PUNCT
ejpam-6813	183	15	)	)	PUNCT
ejpam-6813	183	16	is	be	AUX
ejpam-6813	183	17	a	a	DET
ejpam-6813	183	18	unitarily	unitarily	ADV
ejpam-6813	183	19	invariant	invariant	ADJ
ejpam-6813	183	20	norm	norm	NOUN
ejpam-6813	183	21	.	.	PUNCT
ejpam-6813	184	1	proof	proof	NOUN
ejpam-6813	184	2	.	.	PUNCT
ejpam-6813	185	1	since	since	SCONJ
ejpam-6813	185	2	p	p	NOUN
ejpam-6813	185	3	is	be	AUX
ejpam-6813	185	4	block	block	NOUN
ejpam-6813	185	5	diagonal	diagonal	ADJ
ejpam-6813	185	6	,	,	PUNCT
ejpam-6813	185	7	(	(	PUNCT
ejpam-6813	185	8	pap	pap	ADP
ejpam-6813	185	9	∗)i	∗)i	PROPN
ejpam-6813	185	10	,	,	PUNCT
ejpam-6813	185	11	j	j	PROPN
ejpam-6813	185	12	=	=	SYM
ejpam-6813	185	13	ui	ui	PROPN
ejpam-6813	185	14	,	,	PUNCT
ejpam-6813	185	15	iai	iai	PROPN
ejpam-6813	185	16	,	,	PUNCT
ejpam-6813	185	17	ju	ju	PROPN
ejpam-6813	185	18	∗	∗	PROPN
ejpam-6813	185	19	j	j	PROPN
ejpam-6813	185	20	,	,	PUNCT
ejpam-6813	185	21	j	j	PROPN
ejpam-6813	185	22	for	for	ADP
ejpam-6813	185	23	all	all	DET
ejpam-6813	185	24	i	i	PROPN
ejpam-6813	185	25	,	,	PUNCT
ejpam-6813	185	26	j.	j.	PROPN
ejpam-6813	185	27	by	by	ADP
ejpam-6813	185	28	the	the	DET
ejpam-6813	185	29	unitary	unitary	ADJ
ejpam-6813	185	30	invariance	invariance	NOUN
ejpam-6813	185	31	of	of	ADP
ejpam-6813	185	32	the	the	DET
ejpam-6813	185	33	generalized	generalized	ADJ
ejpam-6813	185	34	numerical	numerical	ADJ
ejpam-6813	185	35	radius	radius	NOUN
ejpam-6813	185	36	:	:	PUNCT
ejpam-6813	185	37	w	w	X
ejpam-6813	185	38	(	(	PUNCT
ejpam-6813	185	39	2	2	NUM
ejpam-6813	185	40	)	)	PUNCT
ejpam-6813	185	41	n	n	CCONJ
ejpam-6813	185	42	(	(	PUNCT
ejpam-6813	185	43	pap	pap	NOUN
ejpam-6813	185	44	∗	∗	NOUN
ejpam-6813	185	45	)	)	PUNCT
ejpam-6813	186	1	=	=	PUNCT
ejpam-6813	187	1	[	[	X
ejpam-6813	187	2	wn	wn	X
ejpam-6813	187	3	(	(	PUNCT
ejpam-6813	187	4	ui	ui	PROPN
ejpam-6813	187	5	,	,	PUNCT
ejpam-6813	187	6	iai	iai	PROPN
ejpam-6813	187	7	,	,	PUNCT
ejpam-6813	187	8	ju	ju	PROPN
ejpam-6813	187	9	∗	∗	PROPN
ejpam-6813	187	10	j	j	PROPN
ejpam-6813	187	11	,	,	PUNCT
ejpam-6813	187	12	j	j	PROPN
ejpam-6813	187	13	)	)	PUNCT
ejpam-6813	187	14	]	]	PUNCT
ejpam-6813	187	15	m	m	VERB
ejpam-6813	187	16	i	i	PRON
ejpam-6813	187	17	,	,	PUNCT
ejpam-6813	187	18	j=1	j=1	PROPN
ejpam-6813	187	19	=	=	PUNCT
ejpam-6813	188	1	[	[	X
ejpam-6813	188	2	wn	wn	X
ejpam-6813	188	3	(	(	PUNCT
ejpam-6813	188	4	ai	ai	PROPN
ejpam-6813	188	5	,	,	PUNCT
ejpam-6813	188	6	j	j	PROPN
ejpam-6813	188	7	)	)	PUNCT
ejpam-6813	188	8	]	]	PUNCT
ejpam-6813	189	1	m	m	VERB
ejpam-6813	189	2	i	i	PRON
ejpam-6813	189	3	,	,	PUNCT
ejpam-6813	189	4	j=1	j=1	PROPN
ejpam-6813	189	5	=	=	PUNCT
ejpam-6813	189	6	w	w	PROPN
ejpam-6813	189	7	(	(	PUNCT
ejpam-6813	189	8	2	2	NUM
ejpam-6813	189	9	)	)	PUNCT
ejpam-6813	189	10	n	n	CCONJ
ejpam-6813	189	11	(	(	PUNCT
ejpam-6813	189	12	a	a	NOUN
ejpam-6813	189	13	)	)	PUNCT
ejpam-6813	189	14	.	.	PUNCT
ejpam-6813	190	1	this	this	PRON
ejpam-6813	190	2	establishes	establish	VERB
ejpam-6813	190	3	that	that	SCONJ
ejpam-6813	190	4	the	the	DET
ejpam-6813	190	5	partial	partial	ADJ
ejpam-6813	190	6	numerical	numerical	ADJ
ejpam-6813	190	7	radius	radius	NOUN
ejpam-6813	190	8	remains	remain	VERB
ejpam-6813	190	9	invariant	invariant	ADJ
ejpam-6813	190	10	under	under	ADP
ejpam-6813	190	11	block	block	NOUN
ejpam-6813	190	12	unitary	unitary	ADJ
ejpam-6813	190	13	similarity	similarity	NOUN
ejpam-6813	190	14	transformations	transformation	NOUN
ejpam-6813	190	15	,	,	PUNCT
ejpam-6813	190	16	extending	extend	VERB
ejpam-6813	190	17	the	the	DET
ejpam-6813	190	18	fundamental	fundamental	ADJ
ejpam-6813	190	19	unitary	unitary	ADJ
ejpam-6813	190	20	invariance	invariance	NOUN
ejpam-6813	190	21	property	property	NOUN
ejpam-6813	190	22	from	from	ADP
ejpam-6813	190	23	individual	individual	ADJ
ejpam-6813	190	24	matrices	matrix	NOUN
ejpam-6813	190	25	to	to	ADP
ejpam-6813	190	26	block	block	NOUN
ejpam-6813	190	27	matrix	matrix	NOUN
ejpam-6813	190	28	structures	structure	NOUN
ejpam-6813	190	29	.	.	PUNCT
ejpam-6813	191	1	4	4	X
ejpam-6813	191	2	.	.	NOUN
ejpam-6813	191	3	block	block	NOUN
ejpam-6813	191	4	diagonal	diagonal	ADJ
ejpam-6813	191	5	matrices	matrix	NOUN
ejpam-6813	191	6	and	and	CCONJ
ejpam-6813	191	7	triangular	triangular	NOUN
ejpam-6813	191	8	structures	structure	NOUN
ejpam-6813	191	9	block	block	VERB
ejpam-6813	191	10	diagonal	diagonal	ADJ
ejpam-6813	191	11	matrices	matrix	NOUN
ejpam-6813	191	12	represent	represent	VERB
ejpam-6813	191	13	one	one	NUM
ejpam-6813	191	14	of	of	ADP
ejpam-6813	191	15	the	the	DET
ejpam-6813	191	16	most	most	ADV
ejpam-6813	191	17	fundamental	fundamental	ADJ
ejpam-6813	191	18	structured	structured	ADJ
ejpam-6813	191	19	matrix	matrix	NOUN
ejpam-6813	191	20	classes	class	NOUN
ejpam-6813	191	21	and	and	CCONJ
ejpam-6813	191	22	provide	provide	VERB
ejpam-6813	191	23	important	important	ADJ
ejpam-6813	191	24	insights	insight	NOUN
ejpam-6813	191	25	into	into	ADP
ejpam-6813	191	26	more	more	ADJ
ejpam-6813	191	27	complex	complex	ADJ
ejpam-6813	191	28	block	block	NOUN
ejpam-6813	191	29	structures	structure	NOUN
ejpam-6813	191	30	.	.	PUNCT
ejpam-6813	192	1	theorem	theorem	NOUN
ejpam-6813	192	2	6	6	NUM
ejpam-6813	192	3	.	.	PUNCT
ejpam-6813	192	4	consider	consider	VERB
ejpam-6813	192	5	a	a	DET
ejpam-6813	192	6	block	block	NOUN
ejpam-6813	192	7	diagonal	diagonal	ADJ
ejpam-6813	192	8	matrix	matrix	NOUN
ejpam-6813	192	9	a	a	DET
ejpam-6813	192	10	=	=	SYM
ejpam-6813	192	11	diag(a1,1	diag(a1,1	PROPN
ejpam-6813	192	12	,	,	PUNCT
ejpam-6813	192	13	a2,2	a2,2	PROPN
ejpam-6813	192	14	,	,	PUNCT
ejpam-6813	192	15	.	.	PUNCT
ejpam-6813	192	16	.	.	PUNCT
ejpam-6813	193	1	.	.	PUNCT
ejpam-6813	194	1	,	,	PUNCT
ejpam-6813	194	2	am	be	AUX
ejpam-6813	194	3	,	,	PUNCT
ejpam-6813	194	4	m	m	NOUN
ejpam-6813	194	5	)	)	PUNCT
ejpam-6813	194	6	∈	∈	PROPN
ejpam-6813	194	7	mm(mn	mm(mn	PROPN
ejpam-6813	194	8	)	)	PUNCT
ejpam-6813	194	9	.	.	PUNCT
ejpam-6813	195	1	then	then	ADV
ejpam-6813	195	2	:	:	PUNCT
ejpam-6813	195	3	w	w	X
ejpam-6813	195	4	(	(	PUNCT
ejpam-6813	195	5	2	2	NUM
ejpam-6813	195	6	)	)	PUNCT
ejpam-6813	195	7	n	n	CCONJ
ejpam-6813	195	8	(	(	PUNCT
ejpam-6813	195	9	a	a	X
ejpam-6813	195	10	)	)	PUNCT
ejpam-6813	195	11	=	=	SYM
ejpam-6813	195	12	diag(wn	diag(wn	X
ejpam-6813	195	13	(	(	PUNCT
ejpam-6813	195	14	a1,1	a1,1	NOUN
ejpam-6813	195	15	)	)	PUNCT
ejpam-6813	195	16	,	,	PUNCT
ejpam-6813	195	17	wn	wn	PROPN
ejpam-6813	195	18	(	(	PUNCT
ejpam-6813	195	19	a2,2	a2,2	PROPN
ejpam-6813	195	20	)	)	PUNCT
ejpam-6813	195	21	,	,	PUNCT
ejpam-6813	195	22	.	.	PUNCT
ejpam-6813	195	23	.	.	PUNCT
ejpam-6813	195	24	.	.	PUNCT
ejpam-6813	196	1	,	,	PUNCT
ejpam-6813	196	2	wn	wn	PROPN
ejpam-6813	196	3	(	(	PUNCT
ejpam-6813	196	4	am	am	PROPN
ejpam-6813	196	5	,	,	PUNCT
ejpam-6813	196	6	m	m	NOUN
ejpam-6813	196	7	)	)	PUNCT
ejpam-6813	196	8	)	)	PUNCT
ejpam-6813	196	9	,	,	PUNCT
ejpam-6813	196	10	where	where	SCONJ
ejpam-6813	196	11	n	n	CCONJ
ejpam-6813	196	12	(	(	PUNCT
ejpam-6813	196	13	·	·	PUNCT
ejpam-6813	196	14	)	)	PUNCT
ejpam-6813	196	15	is	be	AUX
ejpam-6813	196	16	a	a	DET
ejpam-6813	196	17	unitarily	unitarily	ADV
ejpam-6813	196	18	invariant	invariant	ADJ
ejpam-6813	196	19	norm	norm	NOUN
ejpam-6813	196	20	.	.	PUNCT
ejpam-6813	197	1	r.	r.	PROPN
ejpam-6813	197	2	al	al	PROPN
ejpam-6813	197	3	-	-	PROPN
ejpam-6813	197	4	naimi	naimi	PROPN
ejpam-6813	197	5	et	et	PROPN
ejpam-6813	197	6	al	al	PROPN
ejpam-6813	197	7	.	.	PUNCT
ejpam-6813	197	8	/	/	SYM
ejpam-6813	197	9	eur	eur	PROPN
ejpam-6813	197	10	.	.	PUNCT
ejpam-6813	198	1	j.	j.	PROPN
ejpam-6813	198	2	pure	pure	PROPN
ejpam-6813	198	3	appl	appl	PROPN
ejpam-6813	198	4	.	.	PROPN
ejpam-6813	198	5	math	math	PROPN
ejpam-6813	198	6	,	,	PUNCT
ejpam-6813	198	7	18	18	NUM
ejpam-6813	198	8	(	(	PUNCT
ejpam-6813	198	9	4	4	NUM
ejpam-6813	198	10	)	)	PUNCT
ejpam-6813	198	11	(	(	PUNCT
ejpam-6813	198	12	2025	2025	NUM
ejpam-6813	198	13	)	)	PUNCT
ejpam-6813	198	14	,	,	PUNCT
ejpam-6813	198	15	6813	6813	NUM
ejpam-6813	198	16	7	7	NUM
ejpam-6813	198	17	of	of	ADP
ejpam-6813	198	18	15	15	NUM
ejpam-6813	198	19	proof	proof	NOUN
ejpam-6813	198	20	.	.	PUNCT
ejpam-6813	199	1	for	for	ADP
ejpam-6813	199	2	a	a	DET
ejpam-6813	199	3	block	block	NOUN
ejpam-6813	199	4	diagonal	diagonal	ADJ
ejpam-6813	199	5	matrix	matrix	NOUN
ejpam-6813	199	6	a	a	PRON
ejpam-6813	199	7	,	,	PUNCT
ejpam-6813	199	8	we	we	PRON
ejpam-6813	199	9	have	have	AUX
ejpam-6813	199	10	ai	ai	VERB
ejpam-6813	199	11	,	,	PUNCT
ejpam-6813	199	12	j	j	PROPN
ejpam-6813	200	1	=	=	SYM
ejpam-6813	200	2	0	0	PUNCT
ejpam-6813	201	1	whenever	whenever	SCONJ
ejpam-6813	201	2	i	i	PRON
ejpam-6813	201	3	6=	6=	PROPN
ejpam-6813	201	4	j.	j.	PROPN
ejpam-6813	201	5	therefore	therefore	ADV
ejpam-6813	201	6	:	:	PUNCT
ejpam-6813	201	7	w	w	X
ejpam-6813	201	8	(	(	PUNCT
ejpam-6813	201	9	2	2	NUM
ejpam-6813	201	10	)	)	PUNCT
ejpam-6813	201	11	n	n	CCONJ
ejpam-6813	201	12	(	(	PUNCT
ejpam-6813	201	13	a	a	X
ejpam-6813	201	14	)	)	PUNCT
ejpam-6813	201	15	=	=	PUNCT
ejpam-6813	202	1	[	[	X
ejpam-6813	202	2	wn	wn	X
ejpam-6813	202	3	(	(	PUNCT
ejpam-6813	202	4	ai	ai	PROPN
ejpam-6813	202	5	,	,	PUNCT
ejpam-6813	202	6	j	j	PROPN
ejpam-6813	202	7	)	)	PUNCT
ejpam-6813	202	8	]	]	PUNCT
ejpam-6813	203	1	m	m	VERB
ejpam-6813	203	2	i	i	PRON
ejpam-6813	203	3	,	,	PUNCT
ejpam-6813	203	4	j=1	j=1	PROPN
ejpam-6813	203	5	(	(	PUNCT
ejpam-6813	203	6	8)	8)	NUM
ejpam-6813	203	7	=	=	SYM
ejpam-6813	203	8	diag(wn	diag(wn	X
ejpam-6813	203	9	(	(	PUNCT
ejpam-6813	203	10	a1,1	a1,1	NOUN
ejpam-6813	203	11	)	)	PUNCT
ejpam-6813	203	12	,	,	PUNCT
ejpam-6813	203	13	wn	wn	PROPN
ejpam-6813	203	14	(	(	PUNCT
ejpam-6813	203	15	a2,2	a2,2	PROPN
ejpam-6813	203	16	)	)	PUNCT
ejpam-6813	203	17	,	,	PUNCT
ejpam-6813	203	18	.	.	PUNCT
ejpam-6813	203	19	.	.	PUNCT
ejpam-6813	203	20	.	.	PUNCT
ejpam-6813	204	1	,	,	PUNCT
ejpam-6813	204	2	wn	wn	PROPN
ejpam-6813	204	3	(	(	PUNCT
ejpam-6813	204	4	am	am	PROPN
ejpam-6813	204	5	,	,	PUNCT
ejpam-6813	204	6	m	m	NOUN
ejpam-6813	204	7	)	)	PUNCT
ejpam-6813	204	8	)	)	PUNCT
ejpam-6813	204	9	(	(	PUNCT
ejpam-6813	204	10	9	9	X
ejpam-6813	204	11	)	)	PUNCT
ejpam-6813	204	12	since	since	SCONJ
ejpam-6813	204	13	wn	wn	PROPN
ejpam-6813	204	14	(	(	PUNCT
ejpam-6813	204	15	0	0	NUM
ejpam-6813	204	16	)	)	PUNCT
ejpam-6813	204	17	=	=	SYM
ejpam-6813	204	18	0	0	NUM
ejpam-6813	204	19	for	for	ADP
ejpam-6813	204	20	all	all	DET
ejpam-6813	204	21	off	off	ADP
ejpam-6813	204	22	-	-	PUNCT
ejpam-6813	204	23	diagonal	diagonal	ADJ
ejpam-6813	204	24	blocks	block	NOUN
ejpam-6813	204	25	.	.	PUNCT
ejpam-6813	205	1	corollary	corollary	ADJ
ejpam-6813	205	2	2	2	NUM
ejpam-6813	205	3	.	.	X
ejpam-6813	206	1	for	for	ADP
ejpam-6813	206	2	block	block	NOUN
ejpam-6813	206	3	diagonal	diagonal	ADJ
ejpam-6813	206	4	a	a	DET
ejpam-6813	206	5	=	=	SYM
ejpam-6813	206	6	diag(a1,1	diag(a1,1	PROPN
ejpam-6813	206	7	,	,	PUNCT
ejpam-6813	206	8	a2,2	a2,2	PROPN
ejpam-6813	206	9	,	,	PUNCT
ejpam-6813	206	10	.	.	PUNCT
ejpam-6813	206	11	.	.	PUNCT
ejpam-6813	207	1	.	.	PUNCT
ejpam-6813	208	1	,	,	PUNCT
ejpam-6813	208	2	am	be	AUX
ejpam-6813	208	3	,	,	PUNCT
ejpam-6813	208	4	m	m	NOUN
ejpam-6813	208	5	)	)	PUNCT
ejpam-6813	208	6	∈	∈	PROPN
ejpam-6813	208	7	mm(mn	mm(mn	NOUN
ejpam-6813	208	8	):	):	PUNCT
ejpam-6813	208	9	‖w(2)(a)‖	‖w(2)(a)‖	NUM
ejpam-6813	208	10	=	=	SYM
ejpam-6813	208	11	max{w(ai	max{w(ai	PROPN
ejpam-6813	208	12	,	,	PUNCT
ejpam-6813	208	13	i	i	NOUN
ejpam-6813	208	14	)	)	PUNCT
ejpam-6813	208	15	:	:	PUNCT
ejpam-6813	209	1	1	1	NUM
ejpam-6813	209	2	≤	≤	NUM
ejpam-6813	209	3	i	i	X
ejpam-6813	209	4	≤	≤	NUM
ejpam-6813	209	5	m	m	VERB
ejpam-6813	209	6	}	}	PUNCT
ejpam-6813	209	7	upper	upper	ADJ
ejpam-6813	209	8	triangular	triangular	NOUN
ejpam-6813	209	9	block	block	NOUN
ejpam-6813	209	10	matrices	matrix	NOUN
ejpam-6813	209	11	exhibit	exhibit	VERB
ejpam-6813	209	12	rich	rich	ADJ
ejpam-6813	209	13	structural	structural	ADJ
ejpam-6813	209	14	properties	property	NOUN
ejpam-6813	209	15	that	that	PRON
ejpam-6813	209	16	influence	influence	VERB
ejpam-6813	209	17	their	their	PRON
ejpam-6813	209	18	partial	partial	ADJ
ejpam-6813	209	19	numerical	numerical	ADJ
ejpam-6813	209	20	radius	radius	PROPN
ejpam-6813	209	21	behavior	behavior	NOUN
ejpam-6813	209	22	.	.	PUNCT
ejpam-6813	210	1	theorem	theorem	VERB
ejpam-6813	210	2	7	7	NUM
ejpam-6813	210	3	.	.	X
ejpam-6813	210	4	for	for	ADP
ejpam-6813	210	5	upper	upper	ADJ
ejpam-6813	210	6	block	block	NOUN
ejpam-6813	210	7	triangular	triangular	NOUN
ejpam-6813	210	8	a	a	DET
ejpam-6813	210	9	=	=	X
ejpam-6813	210	10	[	[	X
ejpam-6813	210	11	aij	aij	X
ejpam-6813	210	12	]	]	PUNCT
ejpam-6813	210	13	∈	∈	PROPN
ejpam-6813	210	14	mm(mn	mm(mn	PROPN
ejpam-6813	210	15	):	):	PUNCT
ejpam-6813	210	16	max{w(aii	max{w(aii	PROPN
ejpam-6813	210	17	)	)	PUNCT
ejpam-6813	210	18	:	:	PUNCT
ejpam-6813	210	19	1	1	NUM
ejpam-6813	210	20	≤	≤	NUM
ejpam-6813	210	21	i	i	X
ejpam-6813	210	22	≤	≤	NOUN
ejpam-6813	210	23	m	m	VERB
ejpam-6813	210	24	}	}	PUNCT
ejpam-6813	210	25	≤	≤	NOUN
ejpam-6813	210	26	w(a	w(a	PUNCT
ejpam-6813	210	27	)	)	PUNCT
ejpam-6813	210	28	≤	≤	NUM
ejpam-6813	210	29	max	max	PROPN
ejpam-6813	210	30	∑	∑	PROPN
ejpam-6813	210	31	j≥i	j≥i	PROPN
ejpam-6813	210	32	w(aij	w(aij	PROPN
ejpam-6813	210	33	)	)	PUNCT
ejpam-6813	210	34	:	:	PUNCT
ejpam-6813	210	35	1	1	NUM
ejpam-6813	210	36	≤	≤	NUM
ejpam-6813	211	1	i	i	PRON
ejpam-6813	211	2	≤	≤	NOUN
ejpam-6813	211	3	m	m	VERB
ejpam-6813	211	4			NOUN
ejpam-6813	211	5	proof	proof	NOUN
ejpam-6813	211	6	.	.	PUNCT
ejpam-6813	212	1	for	for	ADP
ejpam-6813	212	2	the	the	DET
ejpam-6813	212	3	lower	low	ADJ
ejpam-6813	212	4	bound	bound	ADJ
ejpam-6813	212	5	:	:	PUNCT
ejpam-6813	212	6	let	let	VERB
ejpam-6813	212	7	xi	xi	PRON
ejpam-6813	212	8	be	be	AUX
ejpam-6813	212	9	a	a	DET
ejpam-6813	212	10	unit	unit	NOUN
ejpam-6813	212	11	vector	vector	NOUN
ejpam-6813	212	12	achieving	achieve	VERB
ejpam-6813	212	13	w(aii	w(aii	NOUN
ejpam-6813	212	14	)	)	PUNCT
ejpam-6813	212	15	for	for	ADP
ejpam-6813	212	16	any	any	DET
ejpam-6813	212	17	diagonal	diagonal	ADJ
ejpam-6813	212	18	block	block	NOUN
ejpam-6813	212	19	.	.	PUNCT
ejpam-6813	213	1	define	define	VERB
ejpam-6813	213	2	y	y	PROPN
ejpam-6813	213	3	∈	∈	PROPN
ejpam-6813	213	4	cmn	cmn	NOUN
ejpam-6813	213	5	by	by	ADP
ejpam-6813	213	6	placing	place	VERB
ejpam-6813	213	7	xi	xi	PRON
ejpam-6813	213	8	in	in	ADP
ejpam-6813	213	9	the	the	DET
ejpam-6813	213	10	i	i	PROPN
ejpam-6813	213	11	-	-	PUNCT
ejpam-6813	213	12	th	th	X
ejpam-6813	213	13	block	block	NOUN
ejpam-6813	213	14	position	position	NOUN
ejpam-6813	213	15	and	and	CCONJ
ejpam-6813	213	16	zeros	zero	NOUN
ejpam-6813	213	17	elsewhere	elsewhere	ADV
ejpam-6813	213	18	.	.	PUNCT
ejpam-6813	214	1	then	then	ADV
ejpam-6813	214	2	‖y‖	‖y‖	PROPN
ejpam-6813	214	3	=	=	SYM
ejpam-6813	214	4	1	1	NUM
ejpam-6813	214	5	and	and	CCONJ
ejpam-6813	214	6	:	:	PUNCT
ejpam-6813	214	7	|〈ay	|〈ay	ADJ
ejpam-6813	214	8	,	,	PUNCT
ejpam-6813	214	9	y〉|	y〉|	X
ejpam-6813	214	10	=	=	SYM
ejpam-6813	214	11	|〈aiixi	|〈aiixi	PROPN
ejpam-6813	214	12	,	,	PUNCT
ejpam-6813	214	13	xi〉|	xi〉|	NOUN
ejpam-6813	214	14	=	=	SYM
ejpam-6813	214	15	w(aii	w(aii	PROPN
ejpam-6813	214	16	)	)	PUNCT
ejpam-6813	214	17	therefore	therefore	ADV
ejpam-6813	214	18	w(aii	w(aii	PROPN
ejpam-6813	214	19	)	)	PUNCT
ejpam-6813	214	20	≤	≤	NOUN
ejpam-6813	214	21	w(a	w(a	PUNCT
ejpam-6813	214	22	)	)	PUNCT
ejpam-6813	214	23	for	for	ADP
ejpam-6813	214	24	all	all	DET
ejpam-6813	214	25	i	i	PROPN
ejpam-6813	214	26	,	,	PUNCT
ejpam-6813	214	27	giving	give	VERB
ejpam-6813	214	28	us	we	PRON
ejpam-6813	214	29	the	the	DET
ejpam-6813	214	30	lower	lower	ADV
ejpam-6813	214	31	bound	bind	VERB
ejpam-6813	214	32	.	.	PUNCT
ejpam-6813	215	1	for	for	ADP
ejpam-6813	215	2	the	the	DET
ejpam-6813	215	3	upper	upper	ADJ
ejpam-6813	215	4	bound	bound	NOUN
ejpam-6813	215	5	:	:	PUNCT
ejpam-6813	215	6	for	for	ADP
ejpam-6813	215	7	any	any	DET
ejpam-6813	215	8	unit	unit	NOUN
ejpam-6813	215	9	vector	vector	NOUN
ejpam-6813	215	10	x	x	PUNCT
ejpam-6813	215	11	=	=	PRON
ejpam-6813	215	12	(	(	PUNCT
ejpam-6813	215	13	x1	x1	PROPN
ejpam-6813	215	14	,	,	PUNCT
ejpam-6813	215	15	.	.	PUNCT
ejpam-6813	215	16	.	.	PUNCT
ejpam-6813	215	17	.	.	PUNCT
ejpam-6813	216	1	,	,	PUNCT
ejpam-6813	216	2	xm	xm	PROPN
ejpam-6813	216	3	):	):	PUNCT
ejpam-6813	216	4	|〈ax	|〈ax	NOUN
ejpam-6813	216	5	,	,	PUNCT
ejpam-6813	216	6	x〉|	x〉|	PUNCT
ejpam-6813	216	7	=	=	SYM
ejpam-6813	217	1	∣∣∣∣∣∣	∣∣∣∣∣∣	PROPN
ejpam-6813	217	2	m∑	m∑	CCONJ
ejpam-6813	217	3	i=1	i=1	PROPN
ejpam-6813	217	4	∑	∑	PROPN
ejpam-6813	217	5	j≥i	j≥i	PROPN
ejpam-6813	217	6	〈	〈	PROPN
ejpam-6813	217	7	aijxj	aijxj	NOUN
ejpam-6813	217	8	,	,	PUNCT
ejpam-6813	217	9	xi	xi	PROPN
ejpam-6813	217	10	〉	〉	PROPN
ejpam-6813	217	11	∣∣∣∣∣∣	∣∣∣∣∣∣	PROPN
ejpam-6813	217	12	(	(	PUNCT
ejpam-6813	217	13	10	10	NUM
ejpam-6813	217	14	)	)	PUNCT
ejpam-6813	217	15	≤	≤	NOUN
ejpam-6813	217	16	m∑	m∑	CCONJ
ejpam-6813	217	17	i=1	i=1	PROPN
ejpam-6813	217	18	∑	∑	PROPN
ejpam-6813	217	19	j≥i	j≥i	PROPN
ejpam-6813	217	20	w(aij)‖xj‖‖xi‖	w(aij)‖xj‖‖xi‖	PROPN
ejpam-6813	217	21	(	(	PUNCT
ejpam-6813	217	22	11	11	NUM
ejpam-6813	217	23	)	)	PUNCT
ejpam-6813	217	24	≤	≤	NUM
ejpam-6813	217	25	max	max	PROPN
ejpam-6813	217	26	1≤i≤m	1≤i≤m	NUM
ejpam-6813	217	27	∑	∑	PROPN
ejpam-6813	217	28	j≥i	j≥i	PROPN
ejpam-6813	217	29	w(aij	w(aij	PROPN
ejpam-6813	217	30	)	)	PUNCT
ejpam-6813	218	1			NOUN
ejpam-6813	218	2	m∑	m∑	CCONJ
ejpam-6813	218	3	i=1	i=1	PRON
ejpam-6813	218	4	‖xi‖2	‖xi‖2	X
ejpam-6813	218	5	(	(	PUNCT
ejpam-6813	218	6	12	12	NUM
ejpam-6813	218	7	)	)	PUNCT
ejpam-6813	218	8	=	=	SYM
ejpam-6813	218	9	max	max	PROPN
ejpam-6813	218	10	1≤i≤m	1≤i≤m	NUM
ejpam-6813	218	11	∑	∑	PROPN
ejpam-6813	218	12	j≥i	j≥i	PROPN
ejpam-6813	218	13	w(aij	w(aij	PROPN
ejpam-6813	218	14	)	)	PUNCT
ejpam-6813	219	1			NOUN
ejpam-6813	219	2	(	(	PUNCT
ejpam-6813	219	3	13	13	NUM
ejpam-6813	219	4	)	)	PUNCT
ejpam-6813	219	5	where	where	SCONJ
ejpam-6813	219	6	we	we	PRON
ejpam-6813	219	7	used	use	VERB
ejpam-6813	219	8	∑m	∑m	PROPN
ejpam-6813	219	9	i=1	i=1	PROPN
ejpam-6813	219	10	‖xi‖2	‖xi‖2	X
ejpam-6813	219	11	=	=	SYM
ejpam-6813	219	12	1	1	X
ejpam-6813	219	13	.	.	PUNCT
ejpam-6813	219	14	r.	r.	PROPN
ejpam-6813	219	15	al	al	PROPN
ejpam-6813	219	16	-	-	PROPN
ejpam-6813	219	17	naimi	naimi	PROPN
ejpam-6813	219	18	et	et	PROPN
ejpam-6813	219	19	al	al	PROPN
ejpam-6813	219	20	.	.	PUNCT
ejpam-6813	219	21	/	/	SYM
ejpam-6813	219	22	eur	eur	PROPN
ejpam-6813	219	23	.	.	PUNCT
ejpam-6813	220	1	j.	j.	PROPN
ejpam-6813	220	2	pure	pure	PROPN
ejpam-6813	220	3	appl	appl	PROPN
ejpam-6813	220	4	.	.	PROPN
ejpam-6813	220	5	math	math	PROPN
ejpam-6813	220	6	,	,	PUNCT
ejpam-6813	220	7	18	18	NUM
ejpam-6813	220	8	(	(	PUNCT
ejpam-6813	220	9	4	4	NUM
ejpam-6813	220	10	)	)	PUNCT
ejpam-6813	220	11	(	(	PUNCT
ejpam-6813	220	12	2025	2025	NUM
ejpam-6813	220	13	)	)	PUNCT
ejpam-6813	220	14	,	,	PUNCT
ejpam-6813	220	15	6813	6813	NUM
ejpam-6813	220	16	8	8	NUM
ejpam-6813	220	17	of	of	ADP
ejpam-6813	220	18	15	15	NUM
ejpam-6813	220	19	5	5	NUM
ejpam-6813	220	20	.	.	PUNCT
ejpam-6813	221	1	analysis	analysis	NOUN
ejpam-6813	221	2	of	of	ADP
ejpam-6813	221	3	2×	2×	NUM
ejpam-6813	221	4	2	2	NUM
ejpam-6813	221	5	block	block	NOUN
ejpam-6813	221	6	matrices	matrix	NOUN
ejpam-6813	221	7	the	the	DET
ejpam-6813	221	8	2	2	NUM
ejpam-6813	221	9	×	×	NOUN
ejpam-6813	221	10	2	2	NUM
ejpam-6813	221	11	block	block	NOUN
ejpam-6813	221	12	case	case	NOUN
ejpam-6813	221	13	provides	provide	VERB
ejpam-6813	221	14	crucial	crucial	ADJ
ejpam-6813	221	15	insights	insight	NOUN
ejpam-6813	221	16	that	that	PRON
ejpam-6813	221	17	extend	extend	VERB
ejpam-6813	221	18	to	to	ADP
ejpam-6813	221	19	larger	large	ADJ
ejpam-6813	221	20	block	block	NOUN
ejpam-6813	221	21	matrices	matrix	NOUN
ejpam-6813	221	22	while	while	SCONJ
ejpam-6813	221	23	remaining	remain	VERB
ejpam-6813	221	24	analytically	analytically	ADV
ejpam-6813	221	25	tractable	tractable	ADJ
ejpam-6813	221	26	.	.	PUNCT
ejpam-6813	222	1	theorem	theorem	ADJ
ejpam-6813	222	2	8	8	NUM
ejpam-6813	222	3	.	.	PUNCT
ejpam-6813	223	1	for	for	ADP
ejpam-6813	223	2	a	a	DET
ejpam-6813	223	3	=	=	SYM
ejpam-6813	223	4	[	[	PUNCT
ejpam-6813	223	5	a11	a11	PROPN
ejpam-6813	223	6	a12	a12	PROPN
ejpam-6813	223	7	a21	a21	PROPN
ejpam-6813	223	8	a22	a22	PROPN
ejpam-6813	223	9	]	]	PUNCT
ejpam-6813	223	10	∈	∈	PROPN
ejpam-6813	223	11	m2(mn	m2(mn	PROPN
ejpam-6813	223	12	):	):	PUNCT
ejpam-6813	223	13	max{w(a11	max{w(a11	NOUN
ejpam-6813	223	14	)	)	PUNCT
ejpam-6813	223	15	,	,	PUNCT
ejpam-6813	223	16	w(a22	w(a22	NOUN
ejpam-6813	223	17	)	)	PUNCT
ejpam-6813	223	18	}	}	PUNCT
ejpam-6813	223	19	≤	≤	NOUN
ejpam-6813	223	20	w(a	w(a	PUNCT
ejpam-6813	223	21	)	)	PUNCT
ejpam-6813	223	22	≤	≤	NUM
ejpam-6813	223	23	‖w(2)(a)‖	‖w(2)(a)‖	NUM
ejpam-6813	223	24	proof	proof	NOUN
ejpam-6813	223	25	.	.	PUNCT
ejpam-6813	224	1	for	for	ADP
ejpam-6813	224	2	the	the	DET
ejpam-6813	224	3	lower	low	ADJ
ejpam-6813	224	4	bound	bound	ADJ
ejpam-6813	224	5	:	:	PUNCT
ejpam-6813	224	6	let	let	VERB
ejpam-6813	224	7	x	x	PUNCT
ejpam-6813	224	8	=	=	PRON
ejpam-6813	224	9	(	(	PUNCT
ejpam-6813	224	10	x1	x1	NOUN
ejpam-6813	224	11	0	0	NUM
ejpam-6813	224	12	)	)	PUNCT
ejpam-6813	224	13	where	where	SCONJ
ejpam-6813	224	14	x1	x1	PROPN
ejpam-6813	224	15	maximizes	maximize	VERB
ejpam-6813	224	16	w(a11	w(a11	NOUN
ejpam-6813	224	17	)	)	PUNCT
ejpam-6813	224	18	.	.	PUNCT
ejpam-6813	225	1	then	then	ADV
ejpam-6813	225	2	:	:	PUNCT
ejpam-6813	225	3	|〈ax	|〈ax	ADJ
ejpam-6813	225	4	,	,	PUNCT
ejpam-6813	225	5	x〉|	x〉|	PROPN
ejpam-6813	225	6	=	=	SYM
ejpam-6813	225	7	|〈a11x1	|〈a11x1	NOUN
ejpam-6813	225	8	,	,	PUNCT
ejpam-6813	225	9	x1〉|	x1〉|	PUNCT
ejpam-6813	225	10	=	=	SYM
ejpam-6813	225	11	w(a11	w(a11	NOUN
ejpam-6813	225	12	)	)	PUNCT
ejpam-6813	225	13	therefore	therefore	ADV
ejpam-6813	225	14	w(a	w(a	X
ejpam-6813	225	15	)	)	PUNCT
ejpam-6813	225	16	≥	≥	NOUN
ejpam-6813	225	17	w(a11	w(a11	NOUN
ejpam-6813	225	18	)	)	PUNCT
ejpam-6813	225	19	.	.	PUNCT
ejpam-6813	226	1	similarly	similarly	ADV
ejpam-6813	226	2	,	,	PUNCT
ejpam-6813	226	3	w(a	w(a	PROPN
ejpam-6813	226	4	)	)	PUNCT
ejpam-6813	226	5	≥	≥	NOUN
ejpam-6813	226	6	w(a22	w(a22	NOUN
ejpam-6813	226	7	)	)	PUNCT
ejpam-6813	226	8	.	.	PUNCT
ejpam-6813	227	1	for	for	ADP
ejpam-6813	227	2	the	the	DET
ejpam-6813	227	3	upper	upper	ADJ
ejpam-6813	227	4	bound	bound	NOUN
ejpam-6813	227	5	:	:	PUNCT
ejpam-6813	227	6	for	for	ADP
ejpam-6813	227	7	any	any	DET
ejpam-6813	227	8	unit	unit	NOUN
ejpam-6813	227	9	vector	vector	NOUN
ejpam-6813	227	10	x	x	PUNCT
ejpam-6813	227	11	=	=	PRON
ejpam-6813	227	12	(	(	PUNCT
ejpam-6813	227	13	x1	x1	PROPN
ejpam-6813	227	14	x2	x2	PROPN
ejpam-6813	227	15	)	)	PUNCT
ejpam-6813	227	16	:	:	PUNCT
ejpam-6813	227	17	|〈ax	|〈ax	NOUN
ejpam-6813	227	18	,	,	PUNCT
ejpam-6813	227	19	x〉|	x〉|	PROPN
ejpam-6813	227	20	≤	≤	NUM
ejpam-6813	227	21	w(a11)‖x1‖2	w(a11)‖x1‖2	X
ejpam-6813	227	22	+	+	CCONJ
ejpam-6813	227	23	w(a12)‖x1‖‖x2‖	w(a12)‖x1‖‖x2‖	PROPN
ejpam-6813	227	24	(	(	PUNCT
ejpam-6813	227	25	14	14	NUM
ejpam-6813	227	26	)	)	PUNCT
ejpam-6813	227	27	+	+	CCONJ
ejpam-6813	228	1	w(a21)‖x1‖‖x2‖+	w(a21)‖x1‖‖x2‖+	NOUN
ejpam-6813	228	2	w(a22)‖x2‖2	w(a22)‖x2‖2	NOUN
ejpam-6813	228	3	(	(	PUNCT
ejpam-6813	228	4	15	15	NUM
ejpam-6813	228	5	)	)	PUNCT
ejpam-6813	228	6	let	let	VERB
ejpam-6813	228	7	y	y	NOUN
ejpam-6813	228	8	=	=	PUNCT
ejpam-6813	228	9	(	(	PUNCT
ejpam-6813	228	10	‖x1‖	‖x1‖	NOUN
ejpam-6813	228	11	‖x2‖	‖x2‖	PROPN
ejpam-6813	228	12	)	)	PUNCT
ejpam-6813	228	13	.	.	PUNCT
ejpam-6813	229	1	then	then	ADV
ejpam-6813	229	2	:	:	PUNCT
ejpam-6813	229	3	|〈ax	|〈ax	ADJ
ejpam-6813	229	4	,	,	PUNCT
ejpam-6813	229	5	x〉|	x〉|	PROPN
ejpam-6813	229	6	≤	≤	PUNCT
ejpam-6813	230	1	〈	〈	PROPN
ejpam-6813	230	2	w(2)(a)y	w(2)(a)y	PROPN
ejpam-6813	230	3	,	,	PUNCT
ejpam-6813	230	4	y	y	PROPN
ejpam-6813	230	5	〉	〉	NOUN
ejpam-6813	230	6	≤	≤	NUM
ejpam-6813	230	7	‖w(2)(a)‖	‖w(2)(a)‖	NUM
ejpam-6813	230	8	since	since	SCONJ
ejpam-6813	230	9	‖y‖	‖y‖	PROPN
ejpam-6813	230	10	=	=	SYM
ejpam-6813	230	11	1	1	X
ejpam-6813	230	12	.	.	PUNCT
ejpam-6813	230	13	corollary	corollary	ADJ
ejpam-6813	230	14	3	3	X
ejpam-6813	230	15	.	.	PUNCT
ejpam-6813	231	1	if	if	SCONJ
ejpam-6813	231	2	a	a	PRON
ejpam-6813	231	3	=	=	X
ejpam-6813	231	4	[	[	PUNCT
ejpam-6813	231	5	a11	a11	PROPN
ejpam-6813	231	6	a12	a12	PROPN
ejpam-6813	231	7	a21	a21	PROPN
ejpam-6813	231	8	a22	a22	PROPN
ejpam-6813	231	9	]	]	PUNCT
ejpam-6813	231	10	∈	∈	PROPN
ejpam-6813	231	11	m2(mn	m2(mn	PROPN
ejpam-6813	231	12	)	)	PUNCT
ejpam-6813	231	13	,	,	PUNCT
ejpam-6813	231	14	then	then	ADV
ejpam-6813	231	15	the	the	DET
ejpam-6813	231	16	spectral	spectral	ADJ
ejpam-6813	231	17	norm	norm	NOUN
ejpam-6813	231	18	of	of	ADP
ejpam-6813	231	19	the	the	DET
ejpam-6813	231	20	partial	partial	ADJ
ejpam-6813	231	21	numerical	numerical	ADJ
ejpam-6813	231	22	radius	radius	NOUN
ejpam-6813	231	23	matrix	matrix	NOUN
ejpam-6813	231	24	satisfies	satisfie	NOUN
ejpam-6813	231	25	:	:	PUNCT
ejpam-6813	231	26	‖w(2)(a)‖	‖w(2)(a)‖	NUM
ejpam-6813	231	27	=	=	SYM
ejpam-6813	231	28	1	1	NUM
ejpam-6813	231	29	2	2	NUM
ejpam-6813	231	30	(	(	PUNCT
ejpam-6813	231	31	w(a11	w(a11	NOUN
ejpam-6813	231	32	)	)	PUNCT
ejpam-6813	231	33	+	+	NUM
ejpam-6813	231	34	w(a22	w(a22	NOUN
ejpam-6813	231	35	)	)	PUNCT
ejpam-6813	232	1	+	+	CCONJ
ejpam-6813	232	2	√	√	INTJ
ejpam-6813	232	3	(	(	PUNCT
ejpam-6813	232	4	w(a11)−	w(a11)−	VERB
ejpam-6813	232	5	w(a22))2	w(a22))2	X
ejpam-6813	232	6	+	+	CCONJ
ejpam-6813	232	7	4w(a12)w(a21	4w(a12)w(a21	NUM
ejpam-6813	232	8	)	)	PUNCT
ejpam-6813	232	9	)	)	PUNCT
ejpam-6813	233	1	proof	proof	NOUN
ejpam-6813	233	2	.	.	PUNCT
ejpam-6813	234	1	the	the	DET
ejpam-6813	234	2	matrix	matrix	NOUN
ejpam-6813	234	3	w(2)(a	w(2)(a	NOUN
ejpam-6813	234	4	)	)	PUNCT
ejpam-6813	234	5	=	=	NOUN
ejpam-6813	234	6	[	[	PUNCT
ejpam-6813	234	7	w(a11	w(a11	NOUN
ejpam-6813	234	8	)	)	PUNCT
ejpam-6813	234	9	w(a12	w(a12	ADJ
ejpam-6813	234	10	)	)	PUNCT
ejpam-6813	234	11	w(a21	w(a21	ADJ
ejpam-6813	234	12	)	)	PUNCT
ejpam-6813	234	13	w(a22	w(a22	PROPN
ejpam-6813	234	14	)	)	PUNCT
ejpam-6813	234	15	]	]	PUNCT
ejpam-6813	234	16	has	have	AUX
ejpam-6813	234	17	eigenvalues	eigenvalues	AUX
ejpam-6813	234	18	given	give	VERB
ejpam-6813	234	19	by	by	ADP
ejpam-6813	234	20	the	the	DET
ejpam-6813	234	21	quadratic	quadratic	ADJ
ejpam-6813	234	22	formula	formula	NOUN
ejpam-6813	234	23	.	.	PUNCT
ejpam-6813	235	1	the	the	DET
ejpam-6813	235	2	largest	large	ADJ
ejpam-6813	235	3	eigenvalue	eigenvalue	NOUN
ejpam-6813	235	4	provides	provide	VERB
ejpam-6813	235	5	the	the	DET
ejpam-6813	235	6	spectral	spectral	ADJ
ejpam-6813	235	7	norm	norm	NOUN
ejpam-6813	235	8	.	.	PUNCT
ejpam-6813	236	1	6	6	X
ejpam-6813	236	2	.	.	X
ejpam-6813	236	3	positive	positive	ADJ
ejpam-6813	236	4	semidefinite	semidefinite	NOUN
ejpam-6813	236	5	matrices	matrix	NOUN
ejpam-6813	236	6	and	and	CCONJ
ejpam-6813	236	7	main	main	ADJ
ejpam-6813	236	8	results	result	NOUN
ejpam-6813	236	9	one	one	NUM
ejpam-6813	236	10	of	of	ADP
ejpam-6813	236	11	our	our	PRON
ejpam-6813	236	12	central	central	ADJ
ejpam-6813	236	13	contributions	contribution	NOUN
ejpam-6813	236	14	concerns	concern	VERB
ejpam-6813	236	15	the	the	DET
ejpam-6813	236	16	preservation	preservation	NOUN
ejpam-6813	236	17	of	of	ADP
ejpam-6813	236	18	positive	positive	ADJ
ejpam-6813	236	19	semidefiniteness	semidefiniteness	NOUN
ejpam-6813	236	20	under	under	ADP
ejpam-6813	236	21	partial	partial	ADJ
ejpam-6813	236	22	numerical	numerical	ADJ
ejpam-6813	236	23	radius	radius	NOUN
ejpam-6813	236	24	transformations	transformation	NOUN
ejpam-6813	236	25	.	.	PUNCT
ejpam-6813	237	1	theorem	theorem	NOUN
ejpam-6813	237	2	9	9	NUM
ejpam-6813	237	3	.	.	PUNCT
ejpam-6813	238	1	let	let	VERB
ejpam-6813	238	2	a	a	PRON
ejpam-6813	238	3	=	=	SYM
ejpam-6813	239	1	[	[	X
ejpam-6813	239	2	ai	ai	NOUN
ejpam-6813	239	3	,	,	PUNCT
ejpam-6813	239	4	j	j	PROPN
ejpam-6813	239	5	]	]	PUNCT
ejpam-6813	239	6	∈	∈	PROPN
ejpam-6813	239	7	mm(mn	mm(mn	PROPN
ejpam-6813	239	8	)	)	PUNCT
ejpam-6813	239	9	be	be	AUX
ejpam-6813	239	10	hermitian	hermitian	ADJ
ejpam-6813	239	11	.	.	PUNCT
ejpam-6813	240	1	then	then	ADV
ejpam-6813	240	2	w	w	PROPN
ejpam-6813	240	3	(	(	PUNCT
ejpam-6813	240	4	2	2	NUM
ejpam-6813	240	5	)	)	PUNCT
ejpam-6813	240	6	n	n	CCONJ
ejpam-6813	240	7	(	(	PUNCT
ejpam-6813	240	8	a	a	NOUN
ejpam-6813	240	9	)	)	PUNCT
ejpam-6813	240	10	and	and	CCONJ
ejpam-6813	240	11	w	w	PROPN
ejpam-6813	240	12	(	(	PUNCT
ejpam-6813	240	13	1	1	NUM
ejpam-6813	240	14	)	)	PUNCT
ejpam-6813	240	15	n	n	CCONJ
ejpam-6813	240	16	(	(	PUNCT
ejpam-6813	240	17	a	a	X
ejpam-6813	240	18	)	)	PUNCT
ejpam-6813	240	19	are	be	AUX
ejpam-6813	240	20	symmetric	symmetric	ADJ
ejpam-6813	240	21	and	and	CCONJ
ejpam-6813	240	22	real	real	ADJ
ejpam-6813	240	23	.	.	PUNCT
ejpam-6813	241	1	proof	proof	NOUN
ejpam-6813	241	2	.	.	PUNCT
ejpam-6813	242	1	let	let	VERB
ejpam-6813	242	2	a	a	PRON
ejpam-6813	242	3	=	=	SYM
ejpam-6813	243	1	[	[	X
ejpam-6813	243	2	ai	ai	NOUN
ejpam-6813	243	3	,	,	PUNCT
ejpam-6813	243	4	j	j	PROPN
ejpam-6813	243	5	]	]	PUNCT
ejpam-6813	243	6	∈	∈	PROPN
ejpam-6813	243	7	mm(mn	mm(mn	PROPN
ejpam-6813	243	8	)	)	PUNCT
ejpam-6813	243	9	be	be	AUX
ejpam-6813	243	10	hermitian	hermitian	ADJ
ejpam-6813	243	11	(	(	PUNCT
ejpam-6813	243	12	i.e.	i.e.	X
ejpam-6813	243	13	,	,	PUNCT
ejpam-6813	243	14	a	a	DET
ejpam-6813	243	15	=	=	NOUN
ejpam-6813	243	16	a∗	a∗	NOUN
ejpam-6813	243	17	)	)	PUNCT
ejpam-6813	243	18	.	.	PUNCT
ejpam-6813	244	1	by	by	ADP
ejpam-6813	244	2	theorem	theorem	NOUN
ejpam-6813	244	3	17	17	NUM
ejpam-6813	244	4	,	,	PUNCT
ejpam-6813	244	5	we	we	PRON
ejpam-6813	244	6	have	have	VERB
ejpam-6813	244	7	w	w	NOUN
ejpam-6813	244	8	(	(	PUNCT
ejpam-6813	244	9	2	2	NUM
ejpam-6813	244	10	)	)	PUNCT
ejpam-6813	244	11	n	n	CCONJ
ejpam-6813	244	12	(	(	PUNCT
ejpam-6813	244	13	a	a	X
ejpam-6813	244	14	)	)	PUNCT
ejpam-6813	244	15	=	=	SYM
ejpam-6813	244	16	w	w	PROPN
ejpam-6813	244	17	(	(	PUNCT
ejpam-6813	244	18	2	2	NUM
ejpam-6813	244	19	)	)	PUNCT
ejpam-6813	244	20	n	n	CCONJ
ejpam-6813	244	21	(	(	PUNCT
ejpam-6813	244	22	a∗	a∗	PROPN
ejpam-6813	244	23	)	)	PUNCT
ejpam-6813	244	24	=	=	PUNCT
ejpam-6813	244	25	(	(	PUNCT
ejpam-6813	244	26	w	w	NOUN
ejpam-6813	244	27	(	(	PUNCT
ejpam-6813	244	28	2	2	NUM
ejpam-6813	244	29	)	)	PUNCT
ejpam-6813	244	30	n	n	CCONJ
ejpam-6813	244	31	(	(	PUNCT
ejpam-6813	244	32	a))t	a))t	NOUN
ejpam-6813	244	33	=	=	SYM
ejpam-6813	244	34	(	(	PUNCT
ejpam-6813	244	35	w	w	NOUN
ejpam-6813	244	36	(	(	PUNCT
ejpam-6813	244	37	2	2	NUM
ejpam-6813	244	38	)	)	PUNCT
ejpam-6813	244	39	n	n	PROPN
ejpam-6813	244	40	(	(	PUNCT
ejpam-6813	244	41	a))∗	a))∗	PROPN
ejpam-6813	244	42	,	,	PUNCT
ejpam-6813	244	43	where	where	SCONJ
ejpam-6813	244	44	the	the	DET
ejpam-6813	244	45	third	third	ADJ
ejpam-6813	244	46	equation	equation	NOUN
ejpam-6813	244	47	follows	follow	VERB
ejpam-6813	244	48	because	because	SCONJ
ejpam-6813	244	49	the	the	DET
ejpam-6813	244	50	entries	entry	NOUN
ejpam-6813	244	51	of	of	ADP
ejpam-6813	244	52	w(2	w(2	NOUN
ejpam-6813	244	53	)	)	PUNCT
ejpam-6813	244	54	n	n	CCONJ
ejpam-6813	244	55	(	(	PUNCT
ejpam-6813	244	56	a	a	X
ejpam-6813	244	57	)	)	PUNCT
ejpam-6813	244	58	are	be	AUX
ejpam-6813	244	59	positive	positive	ADJ
ejpam-6813	244	60	real	real	ADJ
ejpam-6813	244	61	numbers	number	NOUN
ejpam-6813	244	62	.	.	PUNCT
ejpam-6813	245	1	therefore	therefore	ADV
ejpam-6813	245	2	w	w	PROPN
ejpam-6813	245	3	(	(	PUNCT
ejpam-6813	245	4	2	2	NUM
ejpam-6813	245	5	)	)	PUNCT
ejpam-6813	245	6	n	n	CCONJ
ejpam-6813	245	7	(	(	PUNCT
ejpam-6813	245	8	a	a	X
ejpam-6813	245	9	)	)	PUNCT
ejpam-6813	245	10	is	be	AUX
ejpam-6813	245	11	hermitian	hermitian	ADJ
ejpam-6813	245	12	.	.	PUNCT
ejpam-6813	246	1	r.	r.	PROPN
ejpam-6813	246	2	al	al	PROPN
ejpam-6813	246	3	-	-	PROPN
ejpam-6813	246	4	naimi	naimi	PROPN
ejpam-6813	246	5	et	et	PROPN
ejpam-6813	246	6	al	al	PROPN
ejpam-6813	246	7	.	.	PUNCT
ejpam-6813	246	8	/	/	SYM
ejpam-6813	246	9	eur	eur	PROPN
ejpam-6813	246	10	.	.	PUNCT
ejpam-6813	247	1	j.	j.	PROPN
ejpam-6813	247	2	pure	pure	PROPN
ejpam-6813	247	3	appl	appl	PROPN
ejpam-6813	247	4	.	.	PROPN
ejpam-6813	247	5	math	math	PROPN
ejpam-6813	247	6	,	,	PUNCT
ejpam-6813	247	7	18	18	NUM
ejpam-6813	247	8	(	(	PUNCT
ejpam-6813	247	9	4	4	NUM
ejpam-6813	247	10	)	)	PUNCT
ejpam-6813	247	11	(	(	PUNCT
ejpam-6813	247	12	2025	2025	NUM
ejpam-6813	247	13	)	)	PUNCT
ejpam-6813	247	14	,	,	PUNCT
ejpam-6813	247	15	6813	6813	NUM
ejpam-6813	247	16	9	9	NUM
ejpam-6813	247	17	of	of	ADP
ejpam-6813	247	18	15	15	NUM
ejpam-6813	247	19	similarly	similarly	ADV
ejpam-6813	247	20	,	,	PUNCT
ejpam-6813	247	21	w(1	w(1	X
ejpam-6813	247	22	)	)	PUNCT
ejpam-6813	247	23	n	n	CCONJ
ejpam-6813	247	24	(	(	PUNCT
ejpam-6813	247	25	a	a	X
ejpam-6813	247	26	)	)	PUNCT
ejpam-6813	247	27	=	=	SYM
ejpam-6813	247	28	w	w	PROPN
ejpam-6813	247	29	(	(	PUNCT
ejpam-6813	247	30	1	1	NUM
ejpam-6813	247	31	)	)	PUNCT
ejpam-6813	247	32	n	n	CCONJ
ejpam-6813	247	33	(	(	PUNCT
ejpam-6813	247	34	a∗	a∗	PROPN
ejpam-6813	247	35	)	)	PUNCT
ejpam-6813	247	36	=	=	PUNCT
ejpam-6813	248	1	(	(	PUNCT
ejpam-6813	248	2	w	w	NOUN
ejpam-6813	248	3	(	(	PUNCT
ejpam-6813	248	4	1	1	NUM
ejpam-6813	248	5	)	)	PUNCT
ejpam-6813	248	6	n	n	CCONJ
ejpam-6813	248	7	(	(	PUNCT
ejpam-6813	248	8	a))t	a))t	NOUN
ejpam-6813	248	9	=	=	SYM
ejpam-6813	248	10	(	(	PUNCT
ejpam-6813	248	11	w	w	NOUN
ejpam-6813	248	12	(	(	PUNCT
ejpam-6813	248	13	1	1	NUM
ejpam-6813	248	14	)	)	PUNCT
ejpam-6813	248	15	n	n	PROPN
ejpam-6813	248	16	(	(	PUNCT
ejpam-6813	248	17	a))∗	a))∗	PROPN
ejpam-6813	248	18	,	,	PUNCT
ejpam-6813	248	19	so	so	SCONJ
ejpam-6813	248	20	w	w	PROPN
ejpam-6813	248	21	(	(	PUNCT
ejpam-6813	248	22	1	1	NUM
ejpam-6813	248	23	)	)	PUNCT
ejpam-6813	248	24	n	n	CCONJ
ejpam-6813	248	25	(	(	PUNCT
ejpam-6813	248	26	a	a	X
ejpam-6813	248	27	)	)	PUNCT
ejpam-6813	248	28	is	be	AUX
ejpam-6813	248	29	hermitian	hermitian	ADJ
ejpam-6813	248	30	.	.	PUNCT
ejpam-6813	249	1	our	our	PRON
ejpam-6813	249	2	main	main	ADJ
ejpam-6813	249	3	theoretical	theoretical	ADJ
ejpam-6813	249	4	contribution	contribution	NOUN
ejpam-6813	249	5	concerns	concern	VERB
ejpam-6813	249	6	the	the	DET
ejpam-6813	249	7	behavior	behavior	NOUN
ejpam-6813	249	8	of	of	ADP
ejpam-6813	249	9	partial	partial	ADJ
ejpam-6813	249	10	numerical	numerical	ADJ
ejpam-6813	249	11	radius	radius	NOUN
ejpam-6813	249	12	for	for	ADP
ejpam-6813	249	13	positive	positive	ADJ
ejpam-6813	249	14	semidefinite	semidefinite	NOUN
ejpam-6813	249	15	matrices	matrix	NOUN
ejpam-6813	249	16	.	.	PUNCT
ejpam-6813	250	1	theorem	theorem	VERB
ejpam-6813	250	2	10	10	NUM
ejpam-6813	250	3	.	.	PUNCT
ejpam-6813	251	1	if	if	SCONJ
ejpam-6813	251	2	a	a	PRON
ejpam-6813	251	3	=	=	PRON
ejpam-6813	252	1	[	[	X
ejpam-6813	252	2	ai	ai	NOUN
ejpam-6813	252	3	,	,	PUNCT
ejpam-6813	252	4	j	j	PROPN
ejpam-6813	252	5	]	]	PUNCT
ejpam-6813	252	6	∈	∈	PROPN
ejpam-6813	252	7	m2(mn	m2(mn	PROPN
ejpam-6813	252	8	)	)	PUNCT
ejpam-6813	252	9	is	be	AUX
ejpam-6813	252	10	positive	positive	ADJ
ejpam-6813	252	11	semidefinite	semidefinite	NOUN
ejpam-6813	252	12	,	,	PUNCT
ejpam-6813	252	13	then	then	ADV
ejpam-6813	252	14	w(2)(a	w(2)(a	NOUN
ejpam-6813	252	15	)	)	PUNCT
ejpam-6813	253	1	is	be	AUX
ejpam-6813	253	2	positive	positive	ADJ
ejpam-6813	253	3	semidefinite	semidefinite	NOUN
ejpam-6813	253	4	.	.	PUNCT
ejpam-6813	254	1	proof	proof	NOUN
ejpam-6813	254	2	.	.	PUNCT
ejpam-6813	255	1	let	let	VERB
ejpam-6813	255	2	a	a	DET
ejpam-6813	255	3	=	=	SYM
ejpam-6813	255	4	[	[	PUNCT
ejpam-6813	255	5	a1,1	a1,1	NOUN
ejpam-6813	255	6	a1,2	a1,2	ADJ
ejpam-6813	255	7	a∗	a∗	PROPN
ejpam-6813	255	8	1,2	1,2	NUM
ejpam-6813	255	9	a2,2	a2,2	PROPN
ejpam-6813	255	10	]	]	PUNCT
ejpam-6813	255	11	be	be	AUX
ejpam-6813	255	12	positive	positive	ADJ
ejpam-6813	255	13	semidefinite	semidefinite	NOUN
ejpam-6813	255	14	.	.	PUNCT
ejpam-6813	256	1	then	then	ADV
ejpam-6813	256	2	for	for	ADP
ejpam-6813	256	3	any	any	DET
ejpam-6813	256	4	z	z	NOUN
ejpam-6813	256	5	,	,	PUNCT
ejpam-6813	256	6	y	y	PROPN
ejpam-6813	256	7	∈	∈	PROPN
ejpam-6813	256	8	cn	cn	NOUN
ejpam-6813	256	9	:	:	PUNCT
ejpam-6813	256	10	|〈a1,2z	|〈a1,2z	PROPN
ejpam-6813	256	11	,	,	PUNCT
ejpam-6813	256	12	y〉|2	y〉|2	PROPN
ejpam-6813	256	13	≤	≤	PUNCT
ejpam-6813	256	14	〈	〈	PROPN
ejpam-6813	256	15	a1,1z	a1,1z	PROPN
ejpam-6813	256	16	,	,	PUNCT
ejpam-6813	256	17	z〉〈a2,2y	z〉〈a2,2y	PROPN
ejpam-6813	256	18	,	,	PUNCT
ejpam-6813	256	19	y	y	PROPN
ejpam-6813	256	20	〉	〉	PROPN
ejpam-6813	256	21	(	(	PUNCT
ejpam-6813	256	22	16	16	NUM
ejpam-6813	256	23	)	)	PUNCT
ejpam-6813	256	24	by	by	ADP
ejpam-6813	256	25	the	the	DET
ejpam-6813	256	26	definition	definition	NOUN
ejpam-6813	256	27	of	of	ADP
ejpam-6813	256	28	numerical	numerical	ADJ
ejpam-6813	256	29	radius	radius	NOUN
ejpam-6813	256	30	:	:	PUNCT
ejpam-6813	256	31	w(a1,2	w(a1,2	NUM
ejpam-6813	256	32	)	)	PUNCT
ejpam-6813	256	33	=	=	SYM
ejpam-6813	256	34	max‖x‖=1	max‖x‖=1	NOUN
ejpam-6813	256	35	|〈a1,2x	|〈a1,2x	PROPN
ejpam-6813	256	36	,	,	PUNCT
ejpam-6813	256	37	x〉|	x〉|	PUNCT
ejpam-6813	256	38	setting	set	VERB
ejpam-6813	256	39	z	z	NOUN
ejpam-6813	256	40	=	=	PUNCT
ejpam-6813	256	41	y	y	PROPN
ejpam-6813	256	42	=	=	PUNCT
ejpam-6813	256	43	v	v	PROPN
ejpam-6813	256	44	where	where	SCONJ
ejpam-6813	256	45	v	v	NOUN
ejpam-6813	256	46	achieves	achieve	VERB
ejpam-6813	256	47	the	the	DET
ejpam-6813	256	48	maximum	maximum	NOUN
ejpam-6813	256	49	in	in	ADP
ejpam-6813	256	50	equation	equation	NOUN
ejpam-6813	256	51	(	(	PUNCT
ejpam-6813	256	52	16	16	NUM
ejpam-6813	256	53	):	):	PUNCT
ejpam-6813	256	54	|〈a1,2v	|〈a1,2v	PROPN
ejpam-6813	256	55	,	,	PUNCT
ejpam-6813	256	56	v〉|2	v〉|2	PROPN
ejpam-6813	256	57	≤	≤	NUM
ejpam-6813	256	58	〈	〈	NOUN
ejpam-6813	256	59	a1,1v	a1,1v	ADP
ejpam-6813	256	60	,	,	PUNCT
ejpam-6813	256	61	v〉〈a2,2v	v〉〈a2,2v	NOUN
ejpam-6813	256	62	,	,	PUNCT
ejpam-6813	256	63	v	v	NOUN
ejpam-6813	256	64	〉	〉	NOUN
ejpam-6813	256	65	≤	≤	NOUN
ejpam-6813	256	66	w(a1,1)w(a2,2	w(a1,1)w(a2,2	PROPN
ejpam-6813	256	67	)	)	PUNCT
ejpam-6813	256	68	hence	hence	ADV
ejpam-6813	256	69	w2(a1,2	w2(a1,2	NOUN
ejpam-6813	256	70	)	)	PUNCT
ejpam-6813	256	71	≤	≤	NOUN
ejpam-6813	256	72	w(a1,1)w(a2,2	w(a1,1)w(a2,2	NUM
ejpam-6813	256	73	)	)	PUNCT
ejpam-6813	256	74	(	(	PUNCT
ejpam-6813	256	75	17	17	NUM
ejpam-6813	256	76	)	)	PUNCT
ejpam-6813	256	77	we	we	PRON
ejpam-6813	256	78	have	have	VERB
ejpam-6813	256	79	w(2)(a	w(2)(a	NOUN
ejpam-6813	256	80	)	)	PUNCT
ejpam-6813	256	81	=	=	PRON
ejpam-6813	257	1	[	[	PUNCT
ejpam-6813	257	2	w(a1,1	w(a1,1	NOUN
ejpam-6813	257	3	)	)	PUNCT
ejpam-6813	257	4	w(a1,2	w(a1,2	VERB
ejpam-6813	257	5	)	)	PUNCT
ejpam-6813	257	6	w(a1,2	w(a1,2	CCONJ
ejpam-6813	257	7	)	)	PUNCT
ejpam-6813	257	8	w(a2,2	w(a2,2	PROPN
ejpam-6813	257	9	)	)	PUNCT
ejpam-6813	257	10	]	]	PUNCT
ejpam-6813	257	11	.	.	PUNCT
ejpam-6813	258	1	by	by	ADP
ejpam-6813	258	2	theorem	theorem	NOUN
ejpam-6813	258	3	9	9	NUM
ejpam-6813	258	4	,	,	PUNCT
ejpam-6813	258	5	w(2)(a	w(2)(a	NOUN
ejpam-6813	258	6	)	)	PUNCT
ejpam-6813	258	7	is	be	AUX
ejpam-6813	258	8	hermitian	hermitian	ADJ
ejpam-6813	258	9	and	and	CCONJ
ejpam-6813	258	10	has	have	VERB
ejpam-6813	258	11	positive	positive	ADJ
ejpam-6813	258	12	diagonal	diagonal	ADJ
ejpam-6813	258	13	entries	entry	NOUN
ejpam-6813	258	14	.	.	PUNCT
ejpam-6813	259	1	for	for	ADP
ejpam-6813	259	2	a	a	DET
ejpam-6813	259	3	2	2	NUM
ejpam-6813	259	4	×	×	NOUN
ejpam-6813	259	5	2	2	NUM
ejpam-6813	259	6	hermitian	hermitian	ADJ
ejpam-6813	259	7	matrix	matrix	NOUN
ejpam-6813	259	8	,	,	PUNCT
ejpam-6813	259	9	positive	positive	ADJ
ejpam-6813	259	10	semidefiniteness	semidefiniteness	NOUN
ejpam-6813	259	11	is	be	AUX
ejpam-6813	259	12	equivalent	equivalent	ADJ
ejpam-6813	259	13	to	to	ADP
ejpam-6813	259	14	non	non	ADJ
ejpam-6813	259	15	-	-	ADJ
ejpam-6813	259	16	negative	negative	ADJ
ejpam-6813	259	17	diagonal	diagonal	ADJ
ejpam-6813	259	18	entries	entry	NOUN
ejpam-6813	259	19	and	and	CCONJ
ejpam-6813	259	20	a	a	DET
ejpam-6813	259	21	non	non	ADJ
ejpam-6813	259	22	-	-	ADJ
ejpam-6813	259	23	negative	negative	ADJ
ejpam-6813	259	24	determinant	determinant	ADJ
ejpam-6813	259	25	.	.	PUNCT
ejpam-6813	260	1	the	the	DET
ejpam-6813	260	2	determinant	determinant	NOUN
ejpam-6813	260	3	of	of	ADP
ejpam-6813	260	4	w(2)(a	w(2)(a	NOUN
ejpam-6813	260	5	)	)	PUNCT
ejpam-6813	260	6	is	be	AUX
ejpam-6813	260	7	:	:	PUNCT
ejpam-6813	260	8	det(w(2)(a	det(w(2)(a	NOUN
ejpam-6813	260	9	)	)	PUNCT
ejpam-6813	260	10	)	)	PUNCT
ejpam-6813	261	1	=	=	PRON
ejpam-6813	261	2	w(a1,1)w(a2,2)−	w(a1,1)w(a2,2)−	VERB
ejpam-6813	261	3	w2(a1,2	w2(a1,2	NOUN
ejpam-6813	261	4	)	)	PUNCT
ejpam-6813	261	5	≥	≥	NOUN
ejpam-6813	261	6	0	0	NUM
ejpam-6813	261	7	by	by	ADP
ejpam-6813	261	8	equation	equation	NOUN
ejpam-6813	261	9	(	(	PUNCT
ejpam-6813	261	10	17	17	NUM
ejpam-6813	261	11	)	)	PUNCT
ejpam-6813	261	12	.	.	PUNCT
ejpam-6813	262	1	therefore	therefore	ADV
ejpam-6813	262	2	w(2)(a	w(2)(a	NOUN
ejpam-6813	262	3	)	)	PUNCT
ejpam-6813	262	4	is	be	AUX
ejpam-6813	262	5	positive	positive	ADJ
ejpam-6813	262	6	semidefinite	semidefinite	NOUN
ejpam-6813	262	7	.	.	PUNCT
ejpam-6813	263	1	theorem	theorem	VERB
ejpam-6813	263	2	11	11	NUM
ejpam-6813	263	3	.	.	PUNCT
ejpam-6813	264	1	if	if	SCONJ
ejpam-6813	264	2	a	a	PRON
ejpam-6813	264	3	=	=	PRON
ejpam-6813	265	1	[	[	X
ejpam-6813	265	2	ai	ai	NOUN
ejpam-6813	265	3	,	,	PUNCT
ejpam-6813	265	4	j	j	PROPN
ejpam-6813	265	5	]	]	X
ejpam-6813	265	6	∈	∈	PROPN
ejpam-6813	265	7	mm(m2	mm(m2	X
ejpam-6813	265	8	)	)	PUNCT
ejpam-6813	265	9	is	be	AUX
ejpam-6813	265	10	positive	positive	ADJ
ejpam-6813	265	11	semidefinite	semidefinite	NOUN
ejpam-6813	265	12	,	,	PUNCT
ejpam-6813	265	13	then	then	ADV
ejpam-6813	265	14	w(1)(a	w(1)(a	NOUN
ejpam-6813	265	15	)	)	PUNCT
ejpam-6813	265	16	is	be	AUX
ejpam-6813	265	17	positive	positive	ADJ
ejpam-6813	265	18	semidefinite	semidefinite	NOUN
ejpam-6813	265	19	.	.	PUNCT
ejpam-6813	266	1	proof	proof	NOUN
ejpam-6813	266	2	.	.	PUNCT
ejpam-6813	267	1	let	let	VERB
ejpam-6813	267	2	a	a	PRON
ejpam-6813	267	3	=	=	SYM
ejpam-6813	268	1	[	[	X
ejpam-6813	268	2	ai	ai	NOUN
ejpam-6813	268	3	,	,	PUNCT
ejpam-6813	268	4	j	j	PROPN
ejpam-6813	268	5	]	]	X
ejpam-6813	268	6	m	m	VERB
ejpam-6813	268	7	i	i	PRON
ejpam-6813	268	8	,	,	PUNCT
ejpam-6813	268	9	j=1	j=1	PROPN
ejpam-6813	268	10	be	be	VERB
ejpam-6813	268	11	positive	positive	ADJ
ejpam-6813	268	12	semidefinite	semidefinite	NOUN
ejpam-6813	268	13	.	.	PUNCT
ejpam-6813	269	1	then	then	ADV
ejpam-6813	269	2	ã	ã	PROPN
ejpam-6813	269	3	=	=	PRON
ejpam-6813	269	4	[	[	PUNCT
ejpam-6813	269	5	g1,1	g1,1	NOUN
ejpam-6813	269	6	g1,2	g1,2	NOUN
ejpam-6813	269	7	g∗	g∗	VERB
ejpam-6813	269	8	1,2	1,2	NUM
ejpam-6813	269	9	g2,2	g2,2	PROPN
ejpam-6813	269	10	]	]	PUNCT
ejpam-6813	269	11	is	be	AUX
ejpam-6813	269	12	positive	positive	ADJ
ejpam-6813	269	13	semidefinite	semidefinite	NOUN
ejpam-6813	269	14	since	since	SCONJ
ejpam-6813	269	15	a	a	PRON
ejpam-6813	269	16	and	and	CCONJ
ejpam-6813	269	17	ã	ã	PROPN
ejpam-6813	269	18	are	be	AUX
ejpam-6813	269	19	unitarily	unitarily	ADV
ejpam-6813	269	20	similar	similar	ADJ
ejpam-6813	269	21	.	.	PUNCT
ejpam-6813	270	1	by	by	ADP
ejpam-6813	270	2	theorem	theorem	ADJ
ejpam-6813	270	3	10	10	NUM
ejpam-6813	270	4	,	,	PUNCT
ejpam-6813	270	5	w(2)(ã	w(2)(ã	PROPN
ejpam-6813	270	6	)	)	PUNCT
ejpam-6813	270	7	is	be	AUX
ejpam-6813	270	8	positive	positive	ADJ
ejpam-6813	270	9	semidefinite	semidefinite	NOUN
ejpam-6813	270	10	.	.	PUNCT
ejpam-6813	271	1	by	by	ADP
ejpam-6813	271	2	remark	remark	NOUN
ejpam-6813	271	3	1	1	NUM
ejpam-6813	271	4	,	,	PUNCT
ejpam-6813	271	5	w(1)(a	w(1)(a	NOUN
ejpam-6813	271	6	)	)	PUNCT
ejpam-6813	271	7	=	=	SYM
ejpam-6813	271	8	w(2)(ã	w(2)(ã	PROPN
ejpam-6813	271	9	)	)	PUNCT
ejpam-6813	271	10	is	be	AUX
ejpam-6813	271	11	positive	positive	ADJ
ejpam-6813	271	12	semidefinite	semidefinite	NOUN
ejpam-6813	271	13	.	.	PUNCT
ejpam-6813	272	1	the	the	DET
ejpam-6813	272	2	following	follow	VERB
ejpam-6813	272	3	example	example	NOUN
ejpam-6813	272	4	demonstrates	demonstrate	VERB
ejpam-6813	272	5	that	that	SCONJ
ejpam-6813	272	6	positive	positive	ADJ
ejpam-6813	272	7	semidefiniteness	semidefiniteness	ADJ
ejpam-6813	272	8	preservation	preservation	NOUN
ejpam-6813	272	9	fails	fail	VERB
ejpam-6813	272	10	for	for	ADP
ejpam-6813	272	11	dimensions	dimension	NOUN
ejpam-6813	272	12	beyond	beyond	ADP
ejpam-6813	272	13	2×	2×	NUM
ejpam-6813	272	14	2	2	NUM
ejpam-6813	272	15	.	.	PUNCT
ejpam-6813	272	16	example	example	NOUN
ejpam-6813	272	17	1	1	NUM
ejpam-6813	272	18	.	.	X
ejpam-6813	272	19	consider	consider	VERB
ejpam-6813	272	20	the	the	DET
ejpam-6813	272	21	3×	3×	NUM
ejpam-6813	272	22	3	3	NUM
ejpam-6813	272	23	matrix	matrix	NOUN
ejpam-6813	272	24	:	:	PUNCT
ejpam-6813	272	25	a	a	DET
ejpam-6813	272	26	=	=	SYM
ejpam-6813	272	27			PROPN
ejpam-6813	272	28	2	2	NUM
ejpam-6813	272	29	−1.9	−1.9	NOUN
ejpam-6813	272	30	0.1	0.1	NUM
ejpam-6813	273	1	−1.9	−1.9	NOUN
ejpam-6813	273	2	2	2	NUM
ejpam-6813	273	3	0	0	NUM
ejpam-6813	273	4	0.1	0.1	NUM
ejpam-6813	273	5	0	0	NUM
ejpam-6813	273	6	2	2	NUM
ejpam-6813	273	7			NOUN
ejpam-6813	273	8	the	the	DET
ejpam-6813	273	9	eigenvalues	eigenvalue	NOUN
ejpam-6813	273	10	of	of	ADP
ejpam-6813	273	11	a	a	PRON
ejpam-6813	273	12	are	be	AUX
ejpam-6813	273	13	approximately	approximately	ADV
ejpam-6813	273	14	:	:	PUNCT
ejpam-6813	274	1	λ1	λ1	PROPN
ejpam-6813	274	2	≈	≈	PROPN
ejpam-6813	274	3	3.9045	3.9045	NUM
ejpam-6813	274	4	,	,	PUNCT
ejpam-6813	274	5	λ2	λ2	PROPN
ejpam-6813	274	6	≈	≈	PROPN
ejpam-6813	274	7	0.0955	0.0955	NUM
ejpam-6813	274	8	,	,	PUNCT
ejpam-6813	274	9	λ3	λ3	PROPN
ejpam-6813	274	10	=	=	PROPN
ejpam-6813	274	11	2.0000	2.0000	NUM
ejpam-6813	274	12	r.	r.	PROPN
ejpam-6813	274	13	al	al	PROPN
ejpam-6813	274	14	-	-	PROPN
ejpam-6813	274	15	naimi	naimi	PROPN
ejpam-6813	274	16	et	et	PROPN
ejpam-6813	274	17	al	al	PROPN
ejpam-6813	274	18	.	.	PUNCT
ejpam-6813	274	19	/	/	SYM
ejpam-6813	274	20	eur	eur	PROPN
ejpam-6813	274	21	.	.	PUNCT
ejpam-6813	275	1	j.	j.	PROPN
ejpam-6813	275	2	pure	pure	PROPN
ejpam-6813	275	3	appl	appl	PROPN
ejpam-6813	275	4	.	.	PROPN
ejpam-6813	275	5	math	math	PROPN
ejpam-6813	275	6	,	,	PUNCT
ejpam-6813	275	7	18	18	NUM
ejpam-6813	275	8	(	(	PUNCT
ejpam-6813	275	9	4	4	NUM
ejpam-6813	275	10	)	)	PUNCT
ejpam-6813	275	11	(	(	PUNCT
ejpam-6813	275	12	2025	2025	NUM
ejpam-6813	275	13	)	)	PUNCT
ejpam-6813	275	14	,	,	PUNCT
ejpam-6813	275	15	6813	6813	NUM
ejpam-6813	275	16	10	10	NUM
ejpam-6813	275	17	of	of	ADP
ejpam-6813	275	18	15	15	NUM
ejpam-6813	275	19	since	since	SCONJ
ejpam-6813	275	20	all	all	DET
ejpam-6813	275	21	eigenvalues	eigenvalue	NOUN
ejpam-6813	275	22	are	be	AUX
ejpam-6813	275	23	positive	positive	ADJ
ejpam-6813	275	24	,	,	PUNCT
ejpam-6813	275	25	a	a	PRON
ejpam-6813	275	26	is	be	AUX
ejpam-6813	275	27	positive	positive	ADJ
ejpam-6813	275	28	semidefinite	semidefinite	NOUN
ejpam-6813	275	29	.	.	PUNCT
ejpam-6813	276	1	however	however	ADV
ejpam-6813	276	2	,	,	PUNCT
ejpam-6813	276	3	for	for	ADP
ejpam-6813	276	4	w(2)(a	w(2)(a	NOUN
ejpam-6813	276	5	)	)	PUNCT
ejpam-6813	276	6	=	=	SYM
ejpam-6813	276	7	w(1)(a	w(1)(a	NOUN
ejpam-6813	276	8	)	)	PUNCT
ejpam-6813	276	9	=	=	PUNCT
ejpam-6813	276	10			NOUN
ejpam-6813	276	11	2	2	NUM
ejpam-6813	276	12	1.9	1.9	NUM
ejpam-6813	276	13	0.1	0.1	NUM
ejpam-6813	276	14	1.9	1.9	NUM
ejpam-6813	276	15	2	2	NUM
ejpam-6813	276	16	0	0	NUM
ejpam-6813	276	17	0.1	0.1	NUM
ejpam-6813	276	18	0	0	NUM
ejpam-6813	276	19	2	2	NUM
ejpam-6813	276	20			NOUN
ejpam-6813	276	21	(	(	PUNCT
ejpam-6813	276	22	using	use	VERB
ejpam-6813	276	23	the	the	DET
ejpam-6813	276	24	standard	standard	ADJ
ejpam-6813	276	25	numerical	numerical	PROPN
ejpam-6813	276	26	radius	radius	PROPN
ejpam-6813	276	27	w	w	PROPN
ejpam-6813	276	28	(	(	PUNCT
ejpam-6813	276	29	·	·	PUNCT
ejpam-6813	276	30	)	)	PUNCT
ejpam-6813	276	31	)	)	PUNCT
ejpam-6813	276	32	,	,	PUNCT
ejpam-6813	276	33	the	the	DET
ejpam-6813	276	34	eigenvalues	eigenvalue	NOUN
ejpam-6813	276	35	are	be	AUX
ejpam-6813	276	36	approximately	approximately	ADV
ejpam-6813	276	37	:	:	PUNCT
ejpam-6813	277	1	λ1	λ1	PROPN
ejpam-6813	277	2	≈	≈	PROPN
ejpam-6813	277	3	4.0045	4.0045	PROPN
ejpam-6813	277	4	,	,	PUNCT
ejpam-6813	277	5	λ2	λ2	PROPN
ejpam-6813	277	6	≈	≈	PROPN
ejpam-6813	277	7	−0.0045	−0.0045	ADJ
ejpam-6813	277	8	,	,	PUNCT
ejpam-6813	277	9	λ3	λ3	PROPN
ejpam-6813	277	10	=	=	PROPN
ejpam-6813	277	11	2.0000	2.0000	NUM
ejpam-6813	277	12	since	since	SCONJ
ejpam-6813	277	13	λ2	λ2	NOUN
ejpam-6813	277	14	<	<	X
ejpam-6813	277	15	0	0	NUM
ejpam-6813	277	16	,	,	PUNCT
ejpam-6813	277	17	w(2)(a	w(2)(a	NOUN
ejpam-6813	277	18	)	)	PUNCT
ejpam-6813	277	19	is	be	AUX
ejpam-6813	277	20	not	not	PART
ejpam-6813	277	21	positive	positive	ADJ
ejpam-6813	277	22	semidefinite	semidefinite	NOUN
ejpam-6813	277	23	,	,	PUNCT
ejpam-6813	277	24	showing	show	VERB
ejpam-6813	277	25	that	that	SCONJ
ejpam-6813	277	26	the	the	DET
ejpam-6813	277	27	property	property	NOUN
ejpam-6813	277	28	fails	fail	VERB
ejpam-6813	277	29	for	for	ADP
ejpam-6813	277	30	n	n	PRON
ejpam-6813	277	31	≥	≥	NOUN
ejpam-6813	277	32	3	3	NUM
ejpam-6813	277	33	.	.	NOUN
ejpam-6813	277	34	7	7	NUM
ejpam-6813	277	35	.	.	PUNCT
ejpam-6813	277	36	additional	additional	ADJ
ejpam-6813	277	37	properties	property	NOUN
ejpam-6813	277	38	and	and	CCONJ
ejpam-6813	277	39	relationships	relationship	NOUN
ejpam-6813	277	40	theorem	theorem	VERB
ejpam-6813	277	41	12	12	NUM
ejpam-6813	277	42	.	.	PUNCT
ejpam-6813	278	1	if	if	SCONJ
ejpam-6813	278	2	a	a	PRON
ejpam-6813	278	3	=	=	PRON
ejpam-6813	279	1	[	[	X
ejpam-6813	279	2	ai	ai	NOUN
ejpam-6813	279	3	,	,	PUNCT
ejpam-6813	279	4	j	j	PROPN
ejpam-6813	279	5	]	]	PUNCT
ejpam-6813	279	6	∈	∈	PROPN
ejpam-6813	279	7	mm(mn	mm(mn	PROPN
ejpam-6813	279	8	)	)	PUNCT
ejpam-6813	279	9	is	be	AUX
ejpam-6813	279	10	partially	partially	ADV
ejpam-6813	279	11	normal	normal	ADJ
ejpam-6813	279	12	,	,	PUNCT
ejpam-6813	279	13	then	then	ADV
ejpam-6813	279	14	r(2)(a	r(2)(a	NOUN
ejpam-6813	279	15	)	)	PUNCT
ejpam-6813	279	16	=	=	PUNCT
ejpam-6813	280	1	‖a‖(2	‖a‖(2	NOUN
ejpam-6813	280	2	)	)	PUNCT
ejpam-6813	280	3	=	=	SYM
ejpam-6813	280	4	w(2)(a	w(2)(a	NOUN
ejpam-6813	280	5	)	)	PUNCT
ejpam-6813	280	6	.	.	PUNCT
ejpam-6813	281	1	proof	proof	NOUN
ejpam-6813	281	2	.	.	PUNCT
ejpam-6813	282	1	since	since	SCONJ
ejpam-6813	282	2	a	a	PRON
ejpam-6813	282	3	is	be	AUX
ejpam-6813	282	4	partially	partially	ADV
ejpam-6813	282	5	normal	normal	ADJ
ejpam-6813	282	6	,	,	PUNCT
ejpam-6813	282	7	ai	ai	VERB
ejpam-6813	282	8	,	,	PUNCT
ejpam-6813	282	9	j	j	PROPN
ejpam-6813	282	10	is	be	AUX
ejpam-6813	282	11	normal	normal	ADJ
ejpam-6813	282	12	for	for	ADP
ejpam-6813	282	13	all	all	DET
ejpam-6813	282	14	1	1	NUM
ejpam-6813	282	15	≤	≤	NUM
ejpam-6813	283	1	i	i	PRON
ejpam-6813	283	2	,	,	PUNCT
ejpam-6813	283	3	j	j	PROPN
ejpam-6813	283	4	≤	≤	PROPN
ejpam-6813	283	5	m.	m.	NOUN
ejpam-6813	283	6	by	by	ADP
ejpam-6813	283	7	relation	relation	NOUN
ejpam-6813	283	8	(	(	PUNCT
ejpam-6813	283	9	3	3	NUM
ejpam-6813	283	10	)	)	PUNCT
ejpam-6813	283	11	,	,	PUNCT
ejpam-6813	283	12	r(ai	r(ai	PROPN
ejpam-6813	283	13	,	,	PUNCT
ejpam-6813	283	14	j	j	NOUN
ejpam-6813	283	15	)	)	PUNCT
ejpam-6813	283	16	=	=	SYM
ejpam-6813	283	17	‖ai	‖ai	NUM
ejpam-6813	283	18	,	,	PUNCT
ejpam-6813	283	19	j‖	j‖	NOUN
ejpam-6813	283	20	=	=	PUNCT
ejpam-6813	283	21	w(ai	w(ai	PROPN
ejpam-6813	283	22	,	,	PUNCT
ejpam-6813	283	23	j	j	NOUN
ejpam-6813	283	24	)	)	PUNCT
ejpam-6813	283	25	for	for	ADP
ejpam-6813	283	26	1	1	NUM
ejpam-6813	283	27	≤	≤	NOUN
ejpam-6813	283	28	i	i	PRON
ejpam-6813	283	29	,	,	PUNCT
ejpam-6813	283	30	j	j	PROPN
ejpam-6813	283	31	≤	≤	PROPN
ejpam-6813	283	32	m.	m.	NOUN
ejpam-6813	283	33	therefore	therefore	ADV
ejpam-6813	283	34	r(2)(a	r(2)(a	NOUN
ejpam-6813	283	35	)	)	PUNCT
ejpam-6813	283	36	=	=	SYM
ejpam-6813	283	37	w(2)(a	w(2)(a	NOUN
ejpam-6813	283	38	)	)	PUNCT
ejpam-6813	283	39	=	=	PUNCT
ejpam-6813	283	40	‖a‖(2	‖a‖(2	NOUN
ejpam-6813	283	41	)	)	PUNCT
ejpam-6813	283	42	.	.	PUNCT
ejpam-6813	284	1	theorem	theorem	NOUN
ejpam-6813	284	2	13	13	NUM
ejpam-6813	284	3	.	.	PUNCT
ejpam-6813	285	1	let	let	VERB
ejpam-6813	285	2	a	a	PRON
ejpam-6813	285	3	=	=	SYM
ejpam-6813	286	1	[	[	X
ejpam-6813	286	2	ai	ai	NOUN
ejpam-6813	286	3	,	,	PUNCT
ejpam-6813	286	4	j	j	PROPN
ejpam-6813	286	5	]	]	PUNCT
ejpam-6813	286	6	∈	∈	PROPN
ejpam-6813	286	7	m2(mn	m2(mn	PROPN
ejpam-6813	286	8	)	)	PUNCT
ejpam-6813	286	9	be	be	AUX
ejpam-6813	286	10	positive	positive	ADJ
ejpam-6813	286	11	semidefinite	semidefinite	NOUN
ejpam-6813	286	12	.	.	PUNCT
ejpam-6813	287	1	then	then	ADV
ejpam-6813	287	2	:	:	PUNCT
ejpam-6813	287	3	2|w(a12)|	2|w(a12)|	NUM
ejpam-6813	287	4	≤	≤	NUM
ejpam-6813	287	5	sj	sj	NOUN
ejpam-6813	287	6	[	[	PUNCT
ejpam-6813	287	7	w(a11	w(a11	NOUN
ejpam-6813	287	8	)	)	PUNCT
ejpam-6813	287	9	w(a12	w(a12	ADJ
ejpam-6813	287	10	)	)	PUNCT
ejpam-6813	287	11	w(a12	w(a12	ADJ
ejpam-6813	287	12	)	)	PUNCT
ejpam-6813	287	13	w(a22	w(a22	NOUN
ejpam-6813	287	14	)	)	PUNCT
ejpam-6813	287	15	]	]	PUNCT
ejpam-6813	288	1	for	for	ADP
ejpam-6813	288	2	j	j	PROPN
ejpam-6813	288	3	=	=	SYM
ejpam-6813	288	4	1	1	NUM
ejpam-6813	288	5	,	,	PUNCT
ejpam-6813	288	6	2	2	NUM
ejpam-6813	288	7	(	(	PUNCT
ejpam-6813	288	8	18	18	NUM
ejpam-6813	288	9	)	)	PUNCT
ejpam-6813	288	10	|w(a12)|	|w(a12)|	PROPN
ejpam-6813	288	11	≤	≤	ADJ
ejpam-6813	288	12	max{w(a11	max{w(a11	NOUN
ejpam-6813	288	13	)	)	PUNCT
ejpam-6813	288	14	,	,	PUNCT
ejpam-6813	288	15	w(a22	w(a22	NOUN
ejpam-6813	288	16	)	)	PUNCT
ejpam-6813	288	17	}	}	PUNCT
ejpam-6813	288	18	(	(	PUNCT
ejpam-6813	288	19	19	19	NUM
ejpam-6813	288	20	)	)	PUNCT
ejpam-6813	288	21	where	where	SCONJ
ejpam-6813	288	22	sj	sj	PROPN
ejpam-6813	288	23	denotes	denote	VERB
ejpam-6813	288	24	the	the	DET
ejpam-6813	288	25	j	j	PROPN
ejpam-6813	288	26	-	-	PUNCT
ejpam-6813	288	27	th	th	VERB
ejpam-6813	288	28	singular	singular	ADJ
ejpam-6813	288	29	value	value	NOUN
ejpam-6813	288	30	.	.	PUNCT
ejpam-6813	289	1	proof	proof	NOUN
ejpam-6813	289	2	.	.	PUNCT
ejpam-6813	290	1	for	for	ADP
ejpam-6813	290	2	a	a	DET
ejpam-6813	290	3	2×2	2×2	NUM
ejpam-6813	290	4	positive	positive	ADJ
ejpam-6813	290	5	semidefinite	semidefinite	NOUN
ejpam-6813	290	6	matrix	matrix	NOUN
ejpam-6813	290	7	[	[	PUNCT
ejpam-6813	290	8	a	a	DET
ejpam-6813	290	9	b	b	PROPN
ejpam-6813	290	10	b	b	PROPN
ejpam-6813	290	11	c	c	PROPN
ejpam-6813	290	12	]	]	PUNCT
ejpam-6813	290	13	,	,	PUNCT
ejpam-6813	290	14	we	we	PRON
ejpam-6813	290	15	have	have	VERB
ejpam-6813	290	16	the	the	DET
ejpam-6813	290	17	classical	classical	ADJ
ejpam-6813	290	18	inequality	inequality	NOUN
ejpam-6813	290	19	|b|	|b|	PROPN
ejpam-6813	290	20	≤	≤	ADV
ejpam-6813	290	21	1	1	NUM
ejpam-6813	290	22	2sj	2sj	NOUN
ejpam-6813	290	23	for	for	ADP
ejpam-6813	290	24	the	the	DET
ejpam-6813	290	25	singular	singular	ADJ
ejpam-6813	290	26	values	value	NOUN
ejpam-6813	290	27	sj	sj	INTJ
ejpam-6813	290	28	(	(	PUNCT
ejpam-6813	290	29	see	see	VERB
ejpam-6813	290	30	[	[	X
ejpam-6813	290	31	2	2	NUM
ejpam-6813	290	32	]	]	NUM
ejpam-6813	290	33	)	)	PUNCT
ejpam-6813	290	34	.	.	PUNCT
ejpam-6813	291	1	applying	apply	VERB
ejpam-6813	291	2	this	this	PRON
ejpam-6813	291	3	to	to	ADP
ejpam-6813	291	4	our	our	PRON
ejpam-6813	291	5	block	block	PROPN
ejpam-6813	291	6	numerical	numerical	ADJ
ejpam-6813	291	7	radius	radius	NOUN
ejpam-6813	291	8	matrix	matrix	NOUN
ejpam-6813	291	9	and	and	CCONJ
ejpam-6813	291	10	using	use	VERB
ejpam-6813	291	11	the	the	DET
ejpam-6813	291	12	fact	fact	NOUN
ejpam-6813	291	13	that	that	SCONJ
ejpam-6813	291	14	for	for	ADP
ejpam-6813	291	15	positive	positive	ADJ
ejpam-6813	291	16	semidefinite	semidefinite	NOUN
ejpam-6813	291	17	matrices	matrix	NOUN
ejpam-6813	291	18	aii	aii	PROPN
ejpam-6813	291	19	,	,	PUNCT
ejpam-6813	291	20	we	we	PRON
ejpam-6813	291	21	have	have	AUX
ejpam-6813	291	22	‖aii‖	‖aii‖	PROPN
ejpam-6813	291	23	=	=	SYM
ejpam-6813	291	24	w(aii	w(aii	PROPN
ejpam-6813	291	25	)	)	PUNCT
ejpam-6813	291	26	,	,	PUNCT
ejpam-6813	291	27	yields	yield	VERB
ejpam-6813	291	28	the	the	DET
ejpam-6813	291	29	desired	desire	VERB
ejpam-6813	291	30	inequalities	inequality	NOUN
ejpam-6813	291	31	.	.	PUNCT
ejpam-6813	292	1	definition	definition	NOUN
ejpam-6813	292	2	4	4	NUM
ejpam-6813	292	3	.	.	PUNCT
ejpam-6813	293	1	if	if	SCONJ
ejpam-6813	293	2	a	a	PRON
ejpam-6813	293	3	=	=	PRON
ejpam-6813	294	1	[	[	X
ejpam-6813	294	2	ai	ai	NOUN
ejpam-6813	294	3	,	,	PUNCT
ejpam-6813	294	4	j	j	PROPN
ejpam-6813	294	5	]	]	PUNCT
ejpam-6813	294	6	∈	∈	PROPN
ejpam-6813	294	7	mm(mn	mm(mn	PROPN
ejpam-6813	294	8	)	)	PUNCT
ejpam-6813	294	9	,	,	PUNCT
ejpam-6813	294	10	then	then	ADV
ejpam-6813	294	11	â	â	X
ejpam-6813	294	12	=	=	PUNCT
ejpam-6813	295	1	[	[	X
ejpam-6813	295	2	a∗	a∗	PROPN
ejpam-6813	295	3	i	i	PROPN
ejpam-6813	295	4	,	,	PUNCT
ejpam-6813	295	5	j	j	PROPN
ejpam-6813	295	6	]	]	X
ejpam-6813	295	7	m	m	VERB
ejpam-6813	295	8	i	i	PRON
ejpam-6813	295	9	,	,	PUNCT
ejpam-6813	295	10	j=1	j=1	PROPN
ejpam-6813	295	11	.	.	PUNCT
ejpam-6813	295	12	remark	remark	PROPN
ejpam-6813	295	13	2	2	NUM
ejpam-6813	295	14	.	.	PUNCT
ejpam-6813	296	1	if	if	SCONJ
ejpam-6813	296	2	a	a	PRON
ejpam-6813	296	3	=	=	PRON
ejpam-6813	297	1	[	[	X
ejpam-6813	297	2	ai	ai	NOUN
ejpam-6813	297	3	,	,	PUNCT
ejpam-6813	297	4	j	j	PROPN
ejpam-6813	297	5	]	]	PUNCT
ejpam-6813	297	6	∈	∈	PROPN
ejpam-6813	297	7	mm(mn	mm(mn	PROPN
ejpam-6813	297	8	)	)	PUNCT
ejpam-6813	297	9	is	be	AUX
ejpam-6813	297	10	partially	partially	ADV
ejpam-6813	297	11	normal	normal	ADJ
ejpam-6813	297	12	,	,	PUNCT
ejpam-6813	297	13	then	then	ADV
ejpam-6813	297	14	ai	ai	VERB
ejpam-6813	297	15	,	,	PUNCT
ejpam-6813	298	1	ja	ja	PROPN
ejpam-6813	298	2	∗	∗	PROPN
ejpam-6813	298	3	i	i	PROPN
ejpam-6813	298	4	,	,	PUNCT
ejpam-6813	298	5	j	j	PROPN
ejpam-6813	298	6	=	=	SYM
ejpam-6813	298	7	a∗	a∗	PROPN
ejpam-6813	298	8	i	i	PROPN
ejpam-6813	298	9	,	,	PUNCT
ejpam-6813	298	10	jai	jai	PROPN
ejpam-6813	298	11	,	,	PUNCT
ejpam-6813	298	12	j	j	PROPN
ejpam-6813	298	13	for	for	ADP
ejpam-6813	298	14	i	i	PROPN
ejpam-6813	298	15	,	,	PUNCT
ejpam-6813	298	16	j	j	PROPN
ejpam-6813	298	17	=	=	SYM
ejpam-6813	298	18	1	1	NUM
ejpam-6813	298	19	,	,	PUNCT
ejpam-6813	298	20	2	2	NUM
ejpam-6813	298	21	,	,	PUNCT
ejpam-6813	298	22	.	.	PUNCT
ejpam-6813	298	23	.	.	PUNCT
ejpam-6813	298	24	.	.	PUNCT
ejpam-6813	299	1	,	,	PUNCT
ejpam-6813	299	2	m.	m.	NOUN
ejpam-6813	299	3	therefore	therefore	ADV
ejpam-6813	299	4	â	â	X
ejpam-6813	299	5	◦	◦	VERB
ejpam-6813	299	6	a	a	DET
ejpam-6813	299	7	=	=	NOUN
ejpam-6813	299	8	a	a	DET
ejpam-6813	299	9	◦	◦	NOUN
ejpam-6813	299	10	â.	â.	ADV
ejpam-6813	299	11	conversely	conversely	ADV
ejpam-6813	299	12	,	,	PUNCT
ejpam-6813	299	13	if	if	SCONJ
ejpam-6813	299	14	â	â	ADP
ejpam-6813	299	15	◦	◦	VERB
ejpam-6813	299	16	a	a	DET
ejpam-6813	299	17	=	=	NOUN
ejpam-6813	299	18	a	a	DET
ejpam-6813	299	19	◦	◦	NOUN
ejpam-6813	299	20	â	â	ADP
ejpam-6813	299	21	,	,	PUNCT
ejpam-6813	299	22	then	then	ADV
ejpam-6813	299	23	ai	ai	VERB
ejpam-6813	299	24	,	,	PUNCT
ejpam-6813	300	1	ja	ja	PROPN
ejpam-6813	300	2	∗	∗	PROPN
ejpam-6813	300	3	i	i	PROPN
ejpam-6813	300	4	,	,	PUNCT
ejpam-6813	300	5	j	j	PROPN
ejpam-6813	300	6	=	=	SYM
ejpam-6813	300	7	a∗	a∗	PROPN
ejpam-6813	300	8	i	i	PROPN
ejpam-6813	300	9	,	,	PUNCT
ejpam-6813	300	10	jai	jai	PROPN
ejpam-6813	300	11	,	,	PUNCT
ejpam-6813	300	12	j	j	PROPN
ejpam-6813	300	13	,	,	PUNCT
ejpam-6813	300	14	so	so	ADV
ejpam-6813	300	15	a	a	PRON
ejpam-6813	300	16	is	be	AUX
ejpam-6813	300	17	partially	partially	ADV
ejpam-6813	300	18	normal	normal	ADJ
ejpam-6813	300	19	.	.	PUNCT
ejpam-6813	301	1	theorem	theorem	VERB
ejpam-6813	301	2	14	14	NUM
ejpam-6813	301	3	.	.	PUNCT
ejpam-6813	302	1	if	if	SCONJ
ejpam-6813	302	2	a	a	PRON
ejpam-6813	302	3	=	=	PRON
ejpam-6813	303	1	[	[	X
ejpam-6813	303	2	ai	ai	NOUN
ejpam-6813	303	3	,	,	PUNCT
ejpam-6813	303	4	j	j	PROPN
ejpam-6813	303	5	]	]	PUNCT
ejpam-6813	303	6	∈	∈	PROPN
ejpam-6813	303	7	mm(mn	mm(mn	PROPN
ejpam-6813	303	8	)	)	PUNCT
ejpam-6813	303	9	,	,	PUNCT
ejpam-6813	303	10	then	then	ADV
ejpam-6813	303	11	w(a	w(a	PROPN
ejpam-6813	303	12	)	)	PUNCT
ejpam-6813	303	13	≤	≤	NUM
ejpam-6813	303	14	min{w(‖a‖(1	min{w(‖a‖(1	PROPN
ejpam-6813	303	15	)	)	PUNCT
ejpam-6813	303	16	)	)	PUNCT
ejpam-6813	303	17	,	,	PUNCT
ejpam-6813	303	18	w(‖a‖(2	w(‖a‖(2	PROPN
ejpam-6813	303	19	)	)	PUNCT
ejpam-6813	303	20	)	)	PUNCT
ejpam-6813	303	21	}	}	PUNCT
ejpam-6813	303	22	.	.	PUNCT
ejpam-6813	304	1	(	(	PUNCT
ejpam-6813	304	2	20	20	X
ejpam-6813	304	3	)	)	PUNCT
ejpam-6813	304	4	proof	proof	NOUN
ejpam-6813	304	5	.	.	PUNCT
ejpam-6813	305	1	since	since	SCONJ
ejpam-6813	305	2	a	a	PRON
ejpam-6813	305	3	and	and	CCONJ
ejpam-6813	305	4	ã	ã	PROPN
ejpam-6813	305	5	are	be	AUX
ejpam-6813	305	6	unitarily	unitarily	ADV
ejpam-6813	305	7	similar	similar	ADJ
ejpam-6813	305	8	,	,	PUNCT
ejpam-6813	305	9	w(a	w(a	PUNCT
ejpam-6813	305	10	)	)	PUNCT
ejpam-6813	305	11	=	=	SYM
ejpam-6813	305	12	w(ã	w(ã	PROPN
ejpam-6813	305	13	)	)	PUNCT
ejpam-6813	305	14	.	.	PUNCT
ejpam-6813	306	1	by	by	ADP
ejpam-6813	306	2	using	use	VERB
ejpam-6813	306	3	inequality	inequality	NOUN
ejpam-6813	306	4	(	(	PUNCT
ejpam-6813	306	5	7	7	NUM
ejpam-6813	306	6	)	)	PUNCT
ejpam-6813	306	7	from	from	ADP
ejpam-6813	306	8	[	[	X
ejpam-6813	306	9	12	12	NUM
ejpam-6813	306	10	]	]	PUNCT
ejpam-6813	306	11	,	,	PUNCT
ejpam-6813	306	12	we	we	PRON
ejpam-6813	306	13	have	have	AUX
ejpam-6813	306	14	w(a	w(a	VERB
ejpam-6813	306	15	)	)	PUNCT
ejpam-6813	306	16	≤	≤	NOUN
ejpam-6813	306	17	w(‖a‖(2	w(‖a‖(2	NUM
ejpam-6813	306	18	)	)	PUNCT
ejpam-6813	306	19	)	)	PUNCT
ejpam-6813	306	20	and	and	CCONJ
ejpam-6813	306	21	w(a	w(a	NUM
ejpam-6813	306	22	)	)	PUNCT
ejpam-6813	306	23	=	=	SYM
ejpam-6813	306	24	w(ã	w(ã	PROPN
ejpam-6813	306	25	)	)	PUNCT
ejpam-6813	306	26	≤	≤	NOUN
ejpam-6813	306	27	w(‖a‖(1	w(‖a‖(1	NOUN
ejpam-6813	306	28	)	)	PUNCT
ejpam-6813	306	29	)	)	PUNCT
ejpam-6813	306	30	.	.	PUNCT
ejpam-6813	307	1	therefore	therefore	ADV
ejpam-6813	307	2	,	,	PUNCT
ejpam-6813	307	3	w(a	w(a	X
ejpam-6813	307	4	)	)	PUNCT
ejpam-6813	307	5	≤	≤	NUM
ejpam-6813	307	6	min{w(‖a‖(1	min{w(‖a‖(1	PROPN
ejpam-6813	307	7	)	)	PUNCT
ejpam-6813	307	8	)	)	PUNCT
ejpam-6813	307	9	,	,	PUNCT
ejpam-6813	307	10	w(‖a‖(2	w(‖a‖(2	PROPN
ejpam-6813	307	11	)	)	PUNCT
ejpam-6813	307	12	)	)	PUNCT
ejpam-6813	307	13	}	}	PUNCT
ejpam-6813	307	14	.	.	PUNCT
ejpam-6813	308	1	r.	r.	PROPN
ejpam-6813	308	2	al	al	PROPN
ejpam-6813	308	3	-	-	PROPN
ejpam-6813	308	4	naimi	naimi	PROPN
ejpam-6813	308	5	et	et	PROPN
ejpam-6813	308	6	al	al	PROPN
ejpam-6813	308	7	.	.	PUNCT
ejpam-6813	308	8	/	/	SYM
ejpam-6813	308	9	eur	eur	PROPN
ejpam-6813	308	10	.	.	PUNCT
ejpam-6813	309	1	j.	j.	PROPN
ejpam-6813	309	2	pure	pure	PROPN
ejpam-6813	309	3	appl	appl	PROPN
ejpam-6813	309	4	.	.	PROPN
ejpam-6813	309	5	math	math	PROPN
ejpam-6813	309	6	,	,	PUNCT
ejpam-6813	309	7	18	18	NUM
ejpam-6813	309	8	(	(	PUNCT
ejpam-6813	309	9	4	4	NUM
ejpam-6813	309	10	)	)	PUNCT
ejpam-6813	309	11	(	(	PUNCT
ejpam-6813	309	12	2025	2025	NUM
ejpam-6813	309	13	)	)	PUNCT
ejpam-6813	309	14	,	,	PUNCT
ejpam-6813	309	15	6813	6813	NUM
ejpam-6813	309	16	11	11	NUM
ejpam-6813	309	17	of	of	ADP
ejpam-6813	309	18	15	15	NUM
ejpam-6813	309	19	theorem	theorem	NOUN
ejpam-6813	309	20	15	15	NUM
ejpam-6813	309	21	.	.	PUNCT
ejpam-6813	310	1	for	for	ADP
ejpam-6813	310	2	a	a	PRON
ejpam-6813	310	3	=	=	PUNCT
ejpam-6813	311	1	[	[	X
ejpam-6813	311	2	ai	ai	NOUN
ejpam-6813	311	3	,	,	PUNCT
ejpam-6813	311	4	j	j	PROPN
ejpam-6813	311	5	]	]	PUNCT
ejpam-6813	311	6	∈	∈	PROPN
ejpam-6813	311	7	mm(mn	mm(mn	PROPN
ejpam-6813	311	8	)	)	PUNCT
ejpam-6813	311	9	,	,	PUNCT
ejpam-6813	311	10	we	we	PRON
ejpam-6813	311	11	have	have	VERB
ejpam-6813	311	12	:	:	PUNCT
ejpam-6813	311	13	tr(w(1)(a	tr(w(1)(a	NUM
ejpam-6813	311	14	)	)	PUNCT
ejpam-6813	311	15	)	)	PUNCT
ejpam-6813	311	16	≤	≤	NUM
ejpam-6813	311	17	tr(‖a‖(1	tr(‖a‖(1	NOUN
ejpam-6813	311	18	)	)	PUNCT
ejpam-6813	311	19	)	)	PUNCT
ejpam-6813	312	1	(	(	PUNCT
ejpam-6813	312	2	21	21	NUM
ejpam-6813	312	3	)	)	PUNCT
ejpam-6813	312	4	tr(w(2)(a	tr(w(2)(a	NUM
ejpam-6813	312	5	)	)	PUNCT
ejpam-6813	312	6	)	)	PUNCT
ejpam-6813	312	7	≤	≤	NOUN
ejpam-6813	312	8	tr(‖a‖(2	tr(‖a‖(2	NOUN
ejpam-6813	312	9	)	)	PUNCT
ejpam-6813	312	10	)	)	PUNCT
ejpam-6813	312	11	(	(	PUNCT
ejpam-6813	312	12	22	22	X
ejpam-6813	312	13	)	)	PUNCT
ejpam-6813	312	14	proof	proof	NOUN
ejpam-6813	312	15	.	.	PUNCT
ejpam-6813	313	1	by	by	ADP
ejpam-6813	313	2	definition	definition	NOUN
ejpam-6813	313	3	of	of	ADP
ejpam-6813	313	4	partial	partial	ADJ
ejpam-6813	313	5	numerical	numerical	ADJ
ejpam-6813	313	6	radius	radius	NOUN
ejpam-6813	313	7	,	,	PUNCT
ejpam-6813	313	8	w(1)(a	w(1)(a	NOUN
ejpam-6813	313	9	)	)	PUNCT
ejpam-6813	313	10	=	=	PUNCT
ejpam-6813	314	1	[	[	X
ejpam-6813	314	2	w(gl	w(gl	NOUN
ejpam-6813	314	3	,	,	PUNCT
ejpam-6813	314	4	k	k	NOUN
ejpam-6813	314	5	)	)	PUNCT
ejpam-6813	314	6	]	]	PUNCT
ejpam-6813	314	7	n	n	PRON
ejpam-6813	314	8	l	l	NOUN
ejpam-6813	314	9	,	,	PUNCT
ejpam-6813	314	10	k=1	k=1	PROPN
ejpam-6813	314	11	.	.	PUNCT
ejpam-6813	315	1	for	for	ADP
ejpam-6813	315	2	any	any	DET
ejpam-6813	315	3	matrix	matrix	NOUN
ejpam-6813	315	4	block	block	NOUN
ejpam-6813	315	5	,	,	PUNCT
ejpam-6813	315	6	w(gl	w(gl	NOUN
ejpam-6813	315	7	,	,	PUNCT
ejpam-6813	315	8	k	k	NOUN
ejpam-6813	315	9	)	)	PUNCT
ejpam-6813	315	10	≤	≤	NOUN
ejpam-6813	315	11	‖gl	‖gl	PROPN
ejpam-6813	315	12	,	,	PUNCT
ejpam-6813	315	13	k‖.	k‖.	NOUN
ejpam-6813	315	14	taking	take	VERB
ejpam-6813	315	15	the	the	DET
ejpam-6813	315	16	trace	trace	NOUN
ejpam-6813	315	17	:	:	PUNCT
ejpam-6813	315	18	tr(w(1)(a	tr(w(1)(a	NUM
ejpam-6813	315	19	)	)	PUNCT
ejpam-6813	315	20	)	)	PUNCT
ejpam-6813	316	1	=	=	PUNCT
ejpam-6813	317	1	n∑	n∑	NOUN
ejpam-6813	317	2	l=1	l=1	NOUN
ejpam-6813	317	3	w(gl	w(gl	NOUN
ejpam-6813	317	4	,	,	PUNCT
ejpam-6813	317	5	l	l	NOUN
ejpam-6813	317	6	)	)	PUNCT
ejpam-6813	317	7	(	(	PUNCT
ejpam-6813	317	8	23	23	NUM
ejpam-6813	317	9	)	)	PUNCT
ejpam-6813	317	10	≤	≤	NOUN
ejpam-6813	318	1	n∑	n∑	PUNCT
ejpam-6813	319	1	l=1	l=1	PROPN
ejpam-6813	320	1	‖gl	‖gl	NUM
ejpam-6813	320	2	,	,	PUNCT
ejpam-6813	320	3	l‖	l‖	NOUN
ejpam-6813	320	4	(	(	PUNCT
ejpam-6813	320	5	24	24	NUM
ejpam-6813	320	6	)	)	PUNCT
ejpam-6813	320	7	=	=	NOUN
ejpam-6813	320	8	tr(‖a‖(1	tr(‖a‖(1	NOUN
ejpam-6813	320	9	)	)	PUNCT
ejpam-6813	320	10	)	)	PUNCT
ejpam-6813	321	1	(	(	PUNCT
ejpam-6813	321	2	25	25	NUM
ejpam-6813	321	3	)	)	PUNCT
ejpam-6813	321	4	the	the	DET
ejpam-6813	321	5	second	second	ADJ
ejpam-6813	321	6	inequality	inequality	NOUN
ejpam-6813	321	7	follows	follow	VERB
ejpam-6813	321	8	by	by	ADP
ejpam-6813	321	9	an	an	DET
ejpam-6813	321	10	analogous	analogous	ADJ
ejpam-6813	321	11	argument	argument	NOUN
ejpam-6813	321	12	.	.	PUNCT
ejpam-6813	322	1	theorem	theorem	VERB
ejpam-6813	322	2	16	16	NUM
ejpam-6813	322	3	.	.	PUNCT
ejpam-6813	323	1	for	for	ADP
ejpam-6813	323	2	a	a	DET
ejpam-6813	323	3	∈	∈	PROPN
ejpam-6813	323	4	mm(mn	mm(mn	NOUN
ejpam-6813	323	5	)	)	PUNCT
ejpam-6813	323	6	,	,	PUNCT
ejpam-6813	323	7	we	we	PRON
ejpam-6813	323	8	have	have	VERB
ejpam-6813	323	9	:	:	PUNCT
ejpam-6813	323	10	tr(w(1)(a	tr(w(1)(a	NUM
ejpam-6813	323	11	)	)	PUNCT
ejpam-6813	323	12	)	)	PUNCT
ejpam-6813	324	1	≤	≤	ADV
ejpam-6813	324	2	1	1	NUM
ejpam-6813	324	3	2	2	NUM
ejpam-6813	324	4	tr(‖a+a∗‖(1	tr(‖a+a∗‖(1	NUM
ejpam-6813	324	5	)	)	PUNCT
ejpam-6813	324	6	)	)	PUNCT
ejpam-6813	325	1	proof	proof	NOUN
ejpam-6813	325	2	.	.	PUNCT
ejpam-6813	326	1	for	for	ADP
ejpam-6813	326	2	each	each	DET
ejpam-6813	326	3	diagonal	diagonal	ADJ
ejpam-6813	326	4	block	block	NOUN
ejpam-6813	326	5	gll	gll	NOUN
ejpam-6813	326	6	,	,	PUNCT
ejpam-6813	326	7	we	we	PRON
ejpam-6813	326	8	have	have	VERB
ejpam-6813	326	9	w(gll	w(gll	NOUN
ejpam-6813	326	10	)	)	PUNCT
ejpam-6813	326	11	≤	≤	NUM
ejpam-6813	326	12	1	1	NUM
ejpam-6813	326	13	2‖gll	2‖gll	NOUN
ejpam-6813	326	14	+	+	CCONJ
ejpam-6813	326	15	g∗	g∗	VERB
ejpam-6813	326	16	ll‖	ll‖	NOUN
ejpam-6813	326	17	by	by	ADP
ejpam-6813	326	18	the	the	DET
ejpam-6813	326	19	classical	classical	ADJ
ejpam-6813	326	20	numerical	numerical	ADJ
ejpam-6813	326	21	radius	radius	PROPN
ejpam-6813	326	22	inequality	inequality	NOUN
ejpam-6813	326	23	.	.	PUNCT
ejpam-6813	327	1	summing	sum	VERB
ejpam-6813	327	2	over	over	ADP
ejpam-6813	327	3	all	all	DET
ejpam-6813	327	4	diagonal	diagonal	ADJ
ejpam-6813	327	5	blocks	block	NOUN
ejpam-6813	327	6	gives	give	VERB
ejpam-6813	327	7	the	the	DET
ejpam-6813	327	8	result	result	NOUN
ejpam-6813	327	9	.	.	PUNCT
ejpam-6813	328	1	theorem	theorem	ADJ
ejpam-6813	328	2	17	17	NUM
ejpam-6813	328	3	.	.	PUNCT
ejpam-6813	329	1	let	let	VERB
ejpam-6813	329	2	a	a	PRON
ejpam-6813	329	3	=	=	SYM
ejpam-6813	330	1	[	[	X
ejpam-6813	330	2	ai	ai	NOUN
ejpam-6813	330	3	,	,	PUNCT
ejpam-6813	330	4	j	j	PROPN
ejpam-6813	330	5	]	]	PUNCT
ejpam-6813	330	6	∈	∈	PROPN
ejpam-6813	330	7	mm(mn	mm(mn	PROPN
ejpam-6813	330	8	)	)	PUNCT
ejpam-6813	330	9	and	and	CCONJ
ejpam-6813	330	10	n	n	CCONJ
ejpam-6813	330	11	be	be	VERB
ejpam-6813	330	12	any	any	DET
ejpam-6813	330	13	self	self	NOUN
ejpam-6813	330	14	-	-	PUNCT
ejpam-6813	330	15	adjoint	adjoint	NOUN
ejpam-6813	330	16	norm	norm	NOUN
ejpam-6813	330	17	.	.	PUNCT
ejpam-6813	331	1	then	then	ADV
ejpam-6813	331	2	:	:	PUNCT
ejpam-6813	331	3	1	1	X
ejpam-6813	331	4	.	.	X
ejpam-6813	331	5	w	w	NOUN
ejpam-6813	331	6	(	(	PUNCT
ejpam-6813	331	7	2	2	NUM
ejpam-6813	331	8	)	)	PUNCT
ejpam-6813	331	9	n	n	NOUN
ejpam-6813	331	10	(	(	PUNCT
ejpam-6813	331	11	aτ	aτ	ADV
ejpam-6813	331	12	)	)	PUNCT
ejpam-6813	331	13	=	=	SYM
ejpam-6813	331	14	(	(	PUNCT
ejpam-6813	331	15	w	w	NOUN
ejpam-6813	331	16	(	(	PUNCT
ejpam-6813	331	17	2	2	NUM
ejpam-6813	331	18	)	)	PUNCT
ejpam-6813	331	19	n	n	CCONJ
ejpam-6813	331	20	(	(	PUNCT
ejpam-6813	331	21	a))t	a))t	PROPN
ejpam-6813	331	22	.	.	PUNCT
ejpam-6813	332	1	2	2	X
ejpam-6813	332	2	.	.	X
ejpam-6813	332	3	w	w	NOUN
ejpam-6813	332	4	(	(	PUNCT
ejpam-6813	332	5	1	1	NUM
ejpam-6813	332	6	)	)	PUNCT
ejpam-6813	332	7	n	n	CCONJ
ejpam-6813	332	8	(	(	PUNCT
ejpam-6813	332	9	a∗	a∗	PROPN
ejpam-6813	332	10	)	)	PUNCT
ejpam-6813	332	11	=	=	PUNCT
ejpam-6813	333	1	(	(	PUNCT
ejpam-6813	333	2	w	w	NOUN
ejpam-6813	333	3	(	(	PUNCT
ejpam-6813	333	4	1	1	NUM
ejpam-6813	333	5	)	)	PUNCT
ejpam-6813	333	6	n	n	CCONJ
ejpam-6813	333	7	(	(	PUNCT
ejpam-6813	333	8	a))t	a))t	NOUN
ejpam-6813	333	9	and	and	CCONJ
ejpam-6813	333	10	w	w	PROPN
ejpam-6813	333	11	(	(	PUNCT
ejpam-6813	333	12	2	2	NUM
ejpam-6813	333	13	)	)	PUNCT
ejpam-6813	333	14	n	n	CCONJ
ejpam-6813	333	15	(	(	PUNCT
ejpam-6813	333	16	a∗	a∗	PROPN
ejpam-6813	333	17	)	)	PUNCT
ejpam-6813	333	18	=	=	PUNCT
ejpam-6813	333	19	(	(	PUNCT
ejpam-6813	333	20	w	w	NOUN
ejpam-6813	333	21	(	(	PUNCT
ejpam-6813	333	22	2	2	NUM
ejpam-6813	333	23	)	)	PUNCT
ejpam-6813	333	24	n	n	CCONJ
ejpam-6813	333	25	(	(	PUNCT
ejpam-6813	333	26	a))t	a))t	PROPN
ejpam-6813	333	27	.	.	PUNCT
ejpam-6813	334	1	3	3	X
ejpam-6813	334	2	.	.	X
ejpam-6813	334	3	w	w	NOUN
ejpam-6813	334	4	(	(	PUNCT
ejpam-6813	334	5	1	1	NUM
ejpam-6813	334	6	)	)	PUNCT
ejpam-6813	334	7	n	n	PROPN
ejpam-6813	334	8	(	(	PUNCT
ejpam-6813	334	9	â	â	PROPN
ejpam-6813	334	10	)	)	PUNCT
ejpam-6813	334	11	=	=	SYM
ejpam-6813	334	12	w	w	PROPN
ejpam-6813	334	13	(	(	PUNCT
ejpam-6813	334	14	1	1	NUM
ejpam-6813	334	15	)	)	PUNCT
ejpam-6813	334	16	n	n	CCONJ
ejpam-6813	334	17	(	(	PUNCT
ejpam-6813	334	18	a	a	NOUN
ejpam-6813	334	19	)	)	PUNCT
ejpam-6813	334	20	and	and	CCONJ
ejpam-6813	334	21	w	w	PROPN
ejpam-6813	334	22	(	(	PUNCT
ejpam-6813	334	23	2	2	NUM
ejpam-6813	334	24	)	)	PUNCT
ejpam-6813	334	25	n	n	PROPN
ejpam-6813	334	26	(	(	PUNCT
ejpam-6813	334	27	â	â	PROPN
ejpam-6813	334	28	)	)	PUNCT
ejpam-6813	334	29	=	=	SYM
ejpam-6813	334	30	w	w	PROPN
ejpam-6813	334	31	(	(	PUNCT
ejpam-6813	334	32	2	2	NUM
ejpam-6813	334	33	)	)	PUNCT
ejpam-6813	334	34	n	n	CCONJ
ejpam-6813	334	35	(	(	PUNCT
ejpam-6813	334	36	a	a	NOUN
ejpam-6813	334	37	)	)	PUNCT
ejpam-6813	334	38	.	.	PUNCT
ejpam-6813	335	1	proof	proof	NOUN
ejpam-6813	335	2	.	.	PUNCT
ejpam-6813	336	1	1	1	X
ejpam-6813	336	2	.	.	X
ejpam-6813	336	3	w	w	NOUN
ejpam-6813	336	4	(	(	PUNCT
ejpam-6813	336	5	2	2	NUM
ejpam-6813	336	6	)	)	PUNCT
ejpam-6813	336	7	n	n	NOUN
ejpam-6813	336	8	(	(	PUNCT
ejpam-6813	336	9	aτ	aτ	ADV
ejpam-6813	336	10	)	)	PUNCT
ejpam-6813	336	11	=	=	PUNCT
ejpam-6813	337	1	[	[	X
ejpam-6813	337	2	wn	wn	X
ejpam-6813	337	3	(	(	PUNCT
ejpam-6813	337	4	aj	aj	PROPN
ejpam-6813	337	5	,	,	PUNCT
ejpam-6813	337	6	i	i	PROPN
ejpam-6813	337	7	)	)	PUNCT
ejpam-6813	337	8	]	]	PUNCT
ejpam-6813	338	1	=	=	PUNCT
ejpam-6813	339	1	[	[	X
ejpam-6813	339	2	wn	wn	X
ejpam-6813	339	3	(	(	PUNCT
ejpam-6813	339	4	ai	ai	PROPN
ejpam-6813	339	5	,	,	PUNCT
ejpam-6813	339	6	j	j	PROPN
ejpam-6813	339	7	)	)	PUNCT
ejpam-6813	339	8	]	]	PUNCT
ejpam-6813	340	1	t	t	PROPN
ejpam-6813	340	2	=	=	SYM
ejpam-6813	340	3	(	(	PUNCT
ejpam-6813	340	4	w	w	NOUN
ejpam-6813	340	5	(	(	PUNCT
ejpam-6813	340	6	2	2	NUM
ejpam-6813	340	7	)	)	PUNCT
ejpam-6813	340	8	n	n	CCONJ
ejpam-6813	340	9	(	(	PUNCT
ejpam-6813	340	10	a))t	a))t	PROPN
ejpam-6813	340	11	.	.	PUNCT
ejpam-6813	341	1	2	2	X
ejpam-6813	341	2	.	.	X
ejpam-6813	341	3	since	since	SCONJ
ejpam-6813	341	4	a∗	a∗	NOUN
ejpam-6813	341	5	=	=	PUNCT
ejpam-6813	341	6	[	[	X
ejpam-6813	341	7	a∗	a∗	PROPN
ejpam-6813	341	8	j	j	PROPN
ejpam-6813	341	9	,	,	PUNCT
ejpam-6813	341	10	i	i	PRON
ejpam-6813	341	11	]	]	X
ejpam-6813	341	12	,	,	PUNCT
ejpam-6813	341	13	we	we	PRON
ejpam-6813	341	14	have	have	VERB
ejpam-6813	341	15	:	:	PUNCT
ejpam-6813	341	16	w	w	X
ejpam-6813	341	17	(	(	PUNCT
ejpam-6813	341	18	1	1	NUM
ejpam-6813	341	19	)	)	PUNCT
ejpam-6813	341	20	n	n	CCONJ
ejpam-6813	341	21	(	(	PUNCT
ejpam-6813	341	22	a∗	a∗	NOUN
ejpam-6813	341	23	)	)	PUNCT
ejpam-6813	341	24	=	=	SYM
ejpam-6813	341	25	w	w	PROPN
ejpam-6813	341	26	(	(	PUNCT
ejpam-6813	341	27	2	2	NUM
ejpam-6813	341	28	)	)	PUNCT
ejpam-6813	341	29	n	n	NOUN
ejpam-6813	341	30	(	(	PUNCT
ejpam-6813	341	31	ã∗	ã∗	PROPN
ejpam-6813	341	32	)	)	PUNCT
ejpam-6813	341	33	=	=	PUNCT
ejpam-6813	342	1	[	[	X
ejpam-6813	342	2	wn	wn	X
ejpam-6813	342	3	(	(	PUNCT
ejpam-6813	342	4	g∗	g∗	PROPN
ejpam-6813	342	5	k	k	PROPN
ejpam-6813	342	6	,	,	PUNCT
ejpam-6813	342	7	l	l	NOUN
ejpam-6813	342	8	)	)	PUNCT
ejpam-6813	342	9	]	]	PUNCT
ejpam-6813	343	1	=	=	PUNCT
ejpam-6813	344	1	[	[	X
ejpam-6813	344	2	wn	wn	X
ejpam-6813	344	3	(	(	PUNCT
ejpam-6813	344	4	gk	gk	PROPN
ejpam-6813	344	5	,	,	PUNCT
ejpam-6813	344	6	l	l	NOUN
ejpam-6813	344	7	)	)	PUNCT
ejpam-6813	344	8	]	]	PUNCT
ejpam-6813	344	9	(	(	PUNCT
ejpam-6813	344	10	26	26	NUM
ejpam-6813	344	11	)	)	PUNCT
ejpam-6813	344	12	=	=	PUNCT
ejpam-6813	345	1	[	[	X
ejpam-6813	345	2	wn	wn	X
ejpam-6813	345	3	(	(	PUNCT
ejpam-6813	345	4	gl	gl	PROPN
ejpam-6813	345	5	,	,	PUNCT
ejpam-6813	345	6	k	k	PROPN
ejpam-6813	345	7	)	)	PUNCT
ejpam-6813	345	8	]	]	PUNCT
ejpam-6813	345	9	t	t	NOUN
ejpam-6813	345	10	=	=	SYM
ejpam-6813	345	11	(	(	PUNCT
ejpam-6813	345	12	w	w	NOUN
ejpam-6813	345	13	(	(	PUNCT
ejpam-6813	345	14	1	1	NUM
ejpam-6813	345	15	)	)	PUNCT
ejpam-6813	345	16	n	n	CCONJ
ejpam-6813	345	17	(	(	PUNCT
ejpam-6813	345	18	a))t	a))t	NOUN
ejpam-6813	345	19	(	(	PUNCT
ejpam-6813	345	20	27	27	NUM
ejpam-6813	345	21	)	)	PUNCT
ejpam-6813	345	22	similarly	similarly	ADV
ejpam-6813	345	23	,	,	PUNCT
ejpam-6813	345	24	w(2	w(2	NOUN
ejpam-6813	345	25	)	)	PUNCT
ejpam-6813	345	26	n	n	PROPN
ejpam-6813	345	27	(	(	PUNCT
ejpam-6813	345	28	a∗	a∗	NOUN
ejpam-6813	345	29	)	)	PUNCT
ejpam-6813	345	30	=	=	PUNCT
ejpam-6813	346	1	[	[	X
ejpam-6813	346	2	wn	wn	X
ejpam-6813	346	3	(	(	PUNCT
ejpam-6813	346	4	a∗	a∗	PROPN
ejpam-6813	346	5	j	j	PROPN
ejpam-6813	346	6	,	,	PUNCT
ejpam-6813	346	7	i	i	PROPN
ejpam-6813	346	8	)	)	PUNCT
ejpam-6813	346	9	]	]	PUNCT
ejpam-6813	347	1	=	=	PUNCT
ejpam-6813	348	1	[	[	X
ejpam-6813	348	2	wn	wn	X
ejpam-6813	348	3	(	(	PUNCT
ejpam-6813	348	4	aj	aj	PROPN
ejpam-6813	348	5	,	,	PUNCT
ejpam-6813	348	6	i	i	PROPN
ejpam-6813	348	7	)	)	PUNCT
ejpam-6813	348	8	]	]	PUNCT
ejpam-6813	349	1	=	=	PUNCT
ejpam-6813	350	1	[	[	X
ejpam-6813	350	2	wn	wn	X
ejpam-6813	350	3	(	(	PUNCT
ejpam-6813	350	4	ai	ai	PROPN
ejpam-6813	350	5	,	,	PUNCT
ejpam-6813	350	6	j	j	PROPN
ejpam-6813	350	7	)	)	PUNCT
ejpam-6813	350	8	]	]	PUNCT
ejpam-6813	351	1	t	t	PROPN
ejpam-6813	351	2	=	=	SYM
ejpam-6813	351	3	(	(	PUNCT
ejpam-6813	351	4	w	w	NOUN
ejpam-6813	351	5	(	(	PUNCT
ejpam-6813	351	6	2	2	NUM
ejpam-6813	351	7	)	)	PUNCT
ejpam-6813	351	8	n	n	PROPN
ejpam-6813	351	9	(	(	PUNCT
ejpam-6813	351	10	a))t	a))t	PROPN
ejpam-6813	351	11	.	.	PUNCT
ejpam-6813	352	1	r.	r.	PROPN
ejpam-6813	352	2	al	al	PROPN
ejpam-6813	352	3	-	-	PROPN
ejpam-6813	352	4	naimi	naimi	PROPN
ejpam-6813	352	5	et	et	PROPN
ejpam-6813	352	6	al	al	PROPN
ejpam-6813	352	7	.	.	PUNCT
ejpam-6813	352	8	/	/	SYM
ejpam-6813	352	9	eur	eur	PROPN
ejpam-6813	352	10	.	.	PUNCT
ejpam-6813	353	1	j.	j.	PROPN
ejpam-6813	353	2	pure	pure	PROPN
ejpam-6813	353	3	appl	appl	PROPN
ejpam-6813	353	4	.	.	PROPN
ejpam-6813	353	5	math	math	PROPN
ejpam-6813	353	6	,	,	PUNCT
ejpam-6813	353	7	18	18	NUM
ejpam-6813	353	8	(	(	PUNCT
ejpam-6813	353	9	4	4	NUM
ejpam-6813	353	10	)	)	PUNCT
ejpam-6813	353	11	(	(	PUNCT
ejpam-6813	353	12	2025	2025	NUM
ejpam-6813	353	13	)	)	PUNCT
ejpam-6813	353	14	,	,	PUNCT
ejpam-6813	353	15	6813	6813	NUM
ejpam-6813	353	16	12	12	NUM
ejpam-6813	353	17	of	of	ADP
ejpam-6813	353	18	15	15	NUM
ejpam-6813	353	19	3	3	NUM
ejpam-6813	353	20	.	.	PUNCT
ejpam-6813	354	1	since	since	SCONJ
ejpam-6813	354	2	â	â	PRON
ejpam-6813	354	3	=	=	PUNCT
ejpam-6813	355	1	[	[	X
ejpam-6813	355	2	a∗	a∗	PROPN
ejpam-6813	355	3	i	i	PROPN
ejpam-6813	355	4	,	,	PUNCT
ejpam-6813	355	5	j	j	PROPN
ejpam-6813	355	6	]	]	PUNCT
ejpam-6813	355	7	,	,	PUNCT
ejpam-6813	355	8	we	we	PRON
ejpam-6813	355	9	have	have	VERB
ejpam-6813	355	10	:	:	PUNCT
ejpam-6813	355	11	w	w	X
ejpam-6813	355	12	(	(	PUNCT
ejpam-6813	355	13	1	1	NUM
ejpam-6813	355	14	)	)	PUNCT
ejpam-6813	355	15	n	n	PROPN
ejpam-6813	355	16	(	(	PUNCT
ejpam-6813	355	17	â	â	PROPN
ejpam-6813	355	18	)	)	PUNCT
ejpam-6813	355	19	=	=	SYM
ejpam-6813	355	20	w	w	PROPN
ejpam-6813	355	21	(	(	PUNCT
ejpam-6813	355	22	2	2	NUM
ejpam-6813	355	23	)	)	PUNCT
ejpam-6813	355	24	n	n	CCONJ
ejpam-6813	355	25	(	(	PUNCT
ejpam-6813	355	26	˜̂	˜̂	X
ejpam-6813	355	27	a	a	X
ejpam-6813	355	28	)	)	PUNCT
ejpam-6813	355	29	=	=	PUNCT
ejpam-6813	356	1	[	[	X
ejpam-6813	356	2	wn	wn	X
ejpam-6813	356	3	(	(	PUNCT
ejpam-6813	356	4	g∗	g∗	PROPN
ejpam-6813	356	5	l	l	PROPN
ejpam-6813	356	6	,	,	PUNCT
ejpam-6813	356	7	k	k	NOUN
ejpam-6813	356	8	)	)	PUNCT
ejpam-6813	356	9	]	]	PUNCT
ejpam-6813	356	10	(	(	PUNCT
ejpam-6813	356	11	28	28	NUM
ejpam-6813	356	12	)	)	PUNCT
ejpam-6813	356	13	=	=	PUNCT
ejpam-6813	357	1	[	[	X
ejpam-6813	357	2	wn	wn	X
ejpam-6813	357	3	(	(	PUNCT
ejpam-6813	357	4	gl	gl	PROPN
ejpam-6813	357	5	,	,	PUNCT
ejpam-6813	357	6	k	k	NOUN
ejpam-6813	357	7	)	)	PUNCT
ejpam-6813	357	8	]	]	PUNCT
ejpam-6813	358	1	=	=	PUNCT
ejpam-6813	358	2	w	w	X
ejpam-6813	358	3	(	(	PUNCT
ejpam-6813	358	4	1	1	NUM
ejpam-6813	358	5	)	)	PUNCT
ejpam-6813	358	6	n	n	CCONJ
ejpam-6813	358	7	(	(	PUNCT
ejpam-6813	358	8	a	a	NOUN
ejpam-6813	358	9	)	)	PUNCT
ejpam-6813	358	10	(	(	PUNCT
ejpam-6813	358	11	29	29	NUM
ejpam-6813	358	12	)	)	PUNCT
ejpam-6813	358	13	similarly	similarly	ADV
ejpam-6813	358	14	,	,	PUNCT
ejpam-6813	358	15	w(2	w(2	NOUN
ejpam-6813	358	16	)	)	PUNCT
ejpam-6813	358	17	n	n	CCONJ
ejpam-6813	358	18	(	(	PUNCT
ejpam-6813	358	19	â	â	ADJ
ejpam-6813	358	20	)	)	PUNCT
ejpam-6813	358	21	=	=	PUNCT
ejpam-6813	359	1	[	[	X
ejpam-6813	359	2	wn	wn	X
ejpam-6813	359	3	(	(	PUNCT
ejpam-6813	359	4	a∗	a∗	PROPN
ejpam-6813	359	5	i	i	PROPN
ejpam-6813	359	6	,	,	PUNCT
ejpam-6813	359	7	j	j	PROPN
ejpam-6813	359	8	)	)	PUNCT
ejpam-6813	359	9	]	]	PUNCT
ejpam-6813	360	1	=	=	PUNCT
ejpam-6813	361	1	[	[	X
ejpam-6813	361	2	wn	wn	X
ejpam-6813	361	3	(	(	PUNCT
ejpam-6813	361	4	ai	ai	PROPN
ejpam-6813	361	5	,	,	PUNCT
ejpam-6813	361	6	j	j	PROPN
ejpam-6813	361	7	)	)	PUNCT
ejpam-6813	361	8	]	]	PUNCT
ejpam-6813	362	1	=	=	PUNCT
ejpam-6813	362	2	w	w	PROPN
ejpam-6813	362	3	(	(	PUNCT
ejpam-6813	362	4	2	2	NUM
ejpam-6813	362	5	)	)	PUNCT
ejpam-6813	362	6	n	n	CCONJ
ejpam-6813	362	7	(	(	PUNCT
ejpam-6813	362	8	a	a	NOUN
ejpam-6813	362	9	)	)	PUNCT
ejpam-6813	362	10	.	.	PUNCT
ejpam-6813	363	1	8	8	X
ejpam-6813	363	2	.	.	PUNCT
ejpam-6813	363	3	enhanced	enhance	VERB
ejpam-6813	363	4	theoretical	theoretical	ADJ
ejpam-6813	363	5	results	result	NOUN
ejpam-6813	363	6	and	and	CCONJ
ejpam-6813	363	7	applications	application	NOUN
ejpam-6813	363	8	8.1	8.1	NUM
ejpam-6813	363	9	.	.	PUNCT
ejpam-6813	364	1	characterization	characterization	NOUN
ejpam-6813	364	2	of	of	ADP
ejpam-6813	364	3	positive	positive	ADJ
ejpam-6813	364	4	semidefinite	semidefinite	NOUN
ejpam-6813	364	5	block	block	NOUN
ejpam-6813	364	6	matrices	matrix	NOUN
ejpam-6813	364	7	theorem	theorem	VERB
ejpam-6813	364	8	18	18	NUM
ejpam-6813	364	9	.	.	PUNCT
ejpam-6813	365	1	for	for	ADP
ejpam-6813	365	2	a	a	PRON
ejpam-6813	365	3	=	=	PUNCT
ejpam-6813	366	1	[	[	X
ejpam-6813	366	2	ai	ai	NOUN
ejpam-6813	366	3	,	,	PUNCT
ejpam-6813	366	4	j	j	PROPN
ejpam-6813	366	5	]	]	PUNCT
ejpam-6813	366	6	∈	∈	PROPN
ejpam-6813	366	7	m2(mn	m2(mn	PROPN
ejpam-6813	366	8	)	)	PUNCT
ejpam-6813	366	9	,	,	PUNCT
ejpam-6813	366	10	the	the	DET
ejpam-6813	366	11	following	follow	VERB
ejpam-6813	366	12	statements	statement	NOUN
ejpam-6813	366	13	are	be	AUX
ejpam-6813	366	14	equivalent	equivalent	ADJ
ejpam-6813	366	15	:	:	PUNCT
ejpam-6813	366	16	(	(	PUNCT
ejpam-6813	366	17	a	a	X
ejpam-6813	366	18	)	)	PUNCT
ejpam-6813	366	19	a	a	PRON
ejpam-6813	366	20	is	be	AUX
ejpam-6813	366	21	positive	positive	ADJ
ejpam-6813	366	22	semidefinite	semidefinite	NOUN
ejpam-6813	366	23	.	.	PUNCT
ejpam-6813	367	1	(	(	PUNCT
ejpam-6813	367	2	b	b	X
ejpam-6813	367	3	)	)	PUNCT
ejpam-6813	367	4	a11	a11	PROPN
ejpam-6813	367	5	,	,	PUNCT
ejpam-6813	367	6	a22	a22	PROPN
ejpam-6813	367	7	≥	≥	NOUN
ejpam-6813	367	8	0	0	NUM
ejpam-6813	367	9	and	and	CCONJ
ejpam-6813	367	10	w2(a12	w2(a12	NUM
ejpam-6813	367	11	)	)	PUNCT
ejpam-6813	367	12	≤	≤	NUM
ejpam-6813	367	13	w(a11)w(a22	w(a11)w(a22	NOUN
ejpam-6813	367	14	)	)	PUNCT
ejpam-6813	367	15	.	.	PUNCT
ejpam-6813	368	1	(	(	PUNCT
ejpam-6813	368	2	c	c	X
ejpam-6813	368	3	)	)	PUNCT
ejpam-6813	368	4	w(2)(a	w(2)(a	NOUN
ejpam-6813	368	5	)	)	PUNCT
ejpam-6813	368	6	is	be	AUX
ejpam-6813	368	7	positive	positive	ADJ
ejpam-6813	368	8	semidefinite	semidefinite	NOUN
ejpam-6813	368	9	.	.	PUNCT
ejpam-6813	369	1	proof	proof	NOUN
ejpam-6813	369	2	.	.	PUNCT
ejpam-6813	370	1	(	(	PUNCT
ejpam-6813	370	2	a	a	X
ejpam-6813	370	3	)	)	PUNCT
ejpam-6813	370	4	⇒	⇒	NOUN
ejpam-6813	370	5	(	(	PUNCT
ejpam-6813	370	6	b	b	X
ejpam-6813	370	7	):	):	PUNCT
ejpam-6813	370	8	if	if	SCONJ
ejpam-6813	370	9	a	a	PRON
ejpam-6813	370	10	is	be	AUX
ejpam-6813	370	11	positive	positive	ADJ
ejpam-6813	370	12	semidefinite	semidefinite	NOUN
ejpam-6813	370	13	,	,	PUNCT
ejpam-6813	370	14	then	then	ADV
ejpam-6813	370	15	clearly	clearly	ADV
ejpam-6813	370	16	a11	a11	PROPN
ejpam-6813	370	17	,	,	PUNCT
ejpam-6813	370	18	a22	a22	PROPN
ejpam-6813	370	19	≥	≥	PROPN
ejpam-6813	370	20	0	0	NUM
ejpam-6813	370	21	.	.	PUNCT
ejpam-6813	371	1	the	the	DET
ejpam-6813	371	2	inequality	inequality	NOUN
ejpam-6813	371	3	follows	follow	VERB
ejpam-6813	371	4	from	from	ADP
ejpam-6813	371	5	the	the	DET
ejpam-6813	371	6	proof	proof	NOUN
ejpam-6813	371	7	of	of	ADP
ejpam-6813	371	8	theorem	theorem	ADJ
ejpam-6813	371	9	10	10	NUM
ejpam-6813	371	10	.	.	PUNCT
ejpam-6813	372	1	(	(	PUNCT
ejpam-6813	372	2	b	b	X
ejpam-6813	372	3	)	)	PUNCT
ejpam-6813	372	4	⇒	⇒	NOUN
ejpam-6813	372	5	(	(	PUNCT
ejpam-6813	372	6	c	c	X
ejpam-6813	372	7	):	):	PUNCT
ejpam-6813	372	8	this	this	PRON
ejpam-6813	372	9	follows	follow	VERB
ejpam-6813	372	10	directly	directly	ADV
ejpam-6813	372	11	from	from	ADP
ejpam-6813	372	12	the	the	DET
ejpam-6813	372	13	proof	proof	NOUN
ejpam-6813	372	14	of	of	ADP
ejpam-6813	372	15	theorem	theorem	ADJ
ejpam-6813	372	16	10	10	NUM
ejpam-6813	372	17	.	.	PUNCT
ejpam-6813	373	1	(	(	PUNCT
ejpam-6813	373	2	c	c	X
ejpam-6813	373	3	)	)	PUNCT
ejpam-6813	373	4	⇒	⇒	NOUN
ejpam-6813	373	5	(	(	PUNCT
ejpam-6813	373	6	a	a	X
ejpam-6813	373	7	):	):	PUNCT
ejpam-6813	373	8	suppose	suppose	VERB
ejpam-6813	373	9	w(2)(a	w(2)(a	NOUN
ejpam-6813	373	10	)	)	PUNCT
ejpam-6813	373	11	is	be	AUX
ejpam-6813	373	12	positive	positive	ADJ
ejpam-6813	373	13	semidefinite	semidefinite	NOUN
ejpam-6813	373	14	.	.	PUNCT
ejpam-6813	374	1	for	for	ADP
ejpam-6813	374	2	any	any	DET
ejpam-6813	374	3	vector	vector	NOUN
ejpam-6813	374	4	v	v	NOUN
ejpam-6813	374	5	=	=	PUNCT
ejpam-6813	374	6	(	(	PUNCT
ejpam-6813	374	7	v1	v1	VERB
ejpam-6813	374	8	v2	v2	PROPN
ejpam-6813	374	9	)	)	PUNCT
ejpam-6813	374	10	:	:	PUNCT
ejpam-6813	374	11	〈	〈	PROPN
ejpam-6813	374	12	av	av	PROPN
ejpam-6813	374	13	,	,	PUNCT
ejpam-6813	374	14	v	v	PROPN
ejpam-6813	374	15	〉	〉	PROPN
ejpam-6813	374	16	≥	≥	NUM
ejpam-6813	374	17	w(a11)‖v1‖2	w(a11)‖v1‖2	NUM
ejpam-6813	374	18	−	−	PROPN
ejpam-6813	374	19	2w(a12)‖v1‖‖v2‖+	2w(a12)‖v1‖‖v2‖+	PROPN
ejpam-6813	374	20	w(a22)‖v2‖2	w(a22)‖v2‖2	PROPN
ejpam-6813	374	21	(	(	PUNCT
ejpam-6813	374	22	30	30	NUM
ejpam-6813	374	23	)	)	PUNCT
ejpam-6813	374	24	since	since	SCONJ
ejpam-6813	374	25	w2(a12	w2(a12	NUM
ejpam-6813	374	26	)	)	PUNCT
ejpam-6813	374	27	≤	≤	NUM
ejpam-6813	374	28	w(a11)w(a22	w(a11)w(a22	NOUN
ejpam-6813	374	29	)	)	PUNCT
ejpam-6813	374	30	,	,	PUNCT
ejpam-6813	374	31	this	this	DET
ejpam-6813	374	32	quadratic	quadratic	ADJ
ejpam-6813	374	33	form	form	NOUN
ejpam-6813	374	34	is	be	AUX
ejpam-6813	374	35	non	non	ADJ
ejpam-6813	374	36	-	-	ADJ
ejpam-6813	374	37	negative	negative	ADJ
ejpam-6813	374	38	,	,	PUNCT
ejpam-6813	374	39	proving	prove	VERB
ejpam-6813	374	40	a	a	DET
ejpam-6813	374	41	≥	≥	NOUN
ejpam-6813	374	42	0	0	NUM
ejpam-6813	374	43	.	.	PUNCT
ejpam-6813	374	44	example	example	NOUN
ejpam-6813	374	45	2	2	NUM
ejpam-6813	374	46	(	(	PUNCT
ejpam-6813	374	47	necessity	necessity	NOUN
ejpam-6813	374	48	of	of	ADP
ejpam-6813	374	49	conditions	condition	NOUN
ejpam-6813	374	50	)	)	PUNCT
ejpam-6813	374	51	.	.	PUNCT
ejpam-6813	375	1	consider	consider	VERB
ejpam-6813	375	2	a	a	DET
ejpam-6813	375	3	=	=	SYM
ejpam-6813	375	4	[	[	PUNCT
ejpam-6813	375	5	1	1	NUM
ejpam-6813	375	6	2	2	NUM
ejpam-6813	375	7	2	2	NUM
ejpam-6813	375	8	1	1	NUM
ejpam-6813	375	9	]	]	PUNCT
ejpam-6813	375	10	⊗	⊗	PROPN
ejpam-6813	375	11	in	in	ADP
ejpam-6813	375	12	.	.	PUNCT
ejpam-6813	376	1	here	here	ADV
ejpam-6813	376	2	w2(a12	w2(a12	NUM
ejpam-6813	376	3	)	)	PUNCT
ejpam-6813	376	4	=	=	PUNCT
ejpam-6813	377	1	4	4	NUM
ejpam-6813	377	2	>	>	SYM
ejpam-6813	377	3	1	1	NUM
ejpam-6813	377	4	=	=	SYM
ejpam-6813	377	5	w(a11)w(a22	w(a11)w(a22	NOUN
ejpam-6813	377	6	)	)	PUNCT
ejpam-6813	377	7	,	,	PUNCT
ejpam-6813	377	8	so	so	CCONJ
ejpam-6813	377	9	condition	condition	NOUN
ejpam-6813	377	10	(	(	PUNCT
ejpam-6813	377	11	b	b	NOUN
ejpam-6813	377	12	)	)	PUNCT
ejpam-6813	377	13	fails	fail	VERB
ejpam-6813	377	14	and	and	CCONJ
ejpam-6813	377	15	a	a	PRON
ejpam-6813	377	16	is	be	AUX
ejpam-6813	377	17	not	not	PART
ejpam-6813	377	18	positive	positive	ADJ
ejpam-6813	377	19	semidefinite	semidefinite	NOUN
ejpam-6813	377	20	.	.	PUNCT
ejpam-6813	378	1	8.2	8.2	NUM
ejpam-6813	378	2	.	.	PUNCT
ejpam-6813	378	3	matrix	matrix	NOUN
ejpam-6813	378	4	function	function	NOUN
ejpam-6813	378	5	analysis	analysis	NOUN
ejpam-6813	378	6	theorem	theorem	VERB
ejpam-6813	378	7	19	19	NUM
ejpam-6813	378	8	.	.	NOUN
ejpam-6813	378	9	for	for	ADP
ejpam-6813	378	10	analytic	analytic	ADJ
ejpam-6813	378	11	function	function	NOUN
ejpam-6813	378	12	f	f	NOUN
ejpam-6813	378	13	and	and	CCONJ
ejpam-6813	378	14	block	block	VERB
ejpam-6813	378	15	diagonal	diagonal	ADJ
ejpam-6813	378	16	a	a	PRON
ejpam-6813	378	17	with	with	ADP
ejpam-6813	378	18	aii	aii	PROPN
ejpam-6813	378	19	≥	≥	NOUN
ejpam-6813	378	20	0	0	NUM
ejpam-6813	378	21	:	:	PUNCT
ejpam-6813	378	22	w(2)(f(a	w(2)(f(a	ADJ
ejpam-6813	378	23	)	)	PUNCT
ejpam-6813	378	24	)	)	PUNCT
ejpam-6813	379	1	≤	≤	NUM
ejpam-6813	379	2	f(w(2)(a	f(w(2)(a	NOUN
ejpam-6813	379	3	)	)	PUNCT
ejpam-6813	379	4	)	)	PUNCT
ejpam-6813	379	5	when	when	SCONJ
ejpam-6813	379	6	f	f	PROPN
ejpam-6813	379	7	is	be	AUX
ejpam-6813	379	8	operator	operator	NOUN
ejpam-6813	379	9	monotone	monotone	NOUN
ejpam-6813	379	10	.	.	PUNCT
ejpam-6813	380	1	proof	proof	NOUN
ejpam-6813	380	2	.	.	PUNCT
ejpam-6813	381	1	for	for	ADP
ejpam-6813	381	2	block	block	NOUN
ejpam-6813	381	3	diagonal	diagonal	ADJ
ejpam-6813	381	4	a	a	DET
ejpam-6813	381	5	=	=	X
ejpam-6813	381	6	diag(a11	diag(a11	PROPN
ejpam-6813	381	7	,	,	PUNCT
ejpam-6813	381	8	.	.	PUNCT
ejpam-6813	381	9	.	.	PUNCT
ejpam-6813	381	10	.	.	PUNCT
ejpam-6813	382	1	,	,	PUNCT
ejpam-6813	382	2	amm	amm	PROPN
ejpam-6813	382	3	)	)	PUNCT
ejpam-6813	382	4	,	,	PUNCT
ejpam-6813	382	5	we	we	PRON
ejpam-6813	382	6	have	have	VERB
ejpam-6813	382	7	f(a	f(a	NOUN
ejpam-6813	382	8	)	)	PUNCT
ejpam-6813	383	1	=	=	SYM
ejpam-6813	383	2	diag(f(a11	diag(f(a11	NOUN
ejpam-6813	383	3	)	)	PUNCT
ejpam-6813	383	4	,	,	PUNCT
ejpam-6813	383	5	.	.	PUNCT
ejpam-6813	383	6	.	.	PUNCT
ejpam-6813	384	1	.	.	PUNCT
ejpam-6813	385	1	,	,	PUNCT
ejpam-6813	385	2	f(amm	f(amm	PROPN
ejpam-6813	385	3	)	)	PUNCT
ejpam-6813	385	4	)	)	PUNCT
ejpam-6813	386	1	and	and	CCONJ
ejpam-6813	386	2	w(2)(a	w(2)(a	NOUN
ejpam-6813	386	3	)	)	PUNCT
ejpam-6813	386	4	=	=	SYM
ejpam-6813	386	5	diag(w(a11	diag(w(a11	NOUN
ejpam-6813	386	6	)	)	PUNCT
ejpam-6813	386	7	,	,	PUNCT
ejpam-6813	386	8	.	.	PUNCT
ejpam-6813	386	9	.	.	PUNCT
ejpam-6813	387	1	.	.	PUNCT
ejpam-6813	388	1	,	,	PUNCT
ejpam-6813	388	2	w(amm	w(amm	PROPN
ejpam-6813	388	3	)	)	PUNCT
ejpam-6813	388	4	)	)	PUNCT
ejpam-6813	388	5	.	.	PUNCT
ejpam-6813	389	1	since	since	SCONJ
ejpam-6813	389	2	f	f	PROPN
ejpam-6813	389	3	is	be	AUX
ejpam-6813	389	4	operator	operator	NOUN
ejpam-6813	389	5	monotone	monotone	NOUN
ejpam-6813	389	6	,	,	PUNCT
ejpam-6813	389	7	w(f(aii	w(f(aii	NOUN
ejpam-6813	389	8	)	)	PUNCT
ejpam-6813	389	9	)	)	PUNCT
ejpam-6813	390	1	≤	≤	NUM
ejpam-6813	390	2	f(w(aii	f(w(aii	NOUN
ejpam-6813	390	3	)	)	PUNCT
ejpam-6813	390	4	)	)	PUNCT
ejpam-6813	390	5	for	for	ADP
ejpam-6813	390	6	each	each	DET
ejpam-6813	390	7	i.	i.	NOUN
ejpam-6813	390	8	therefore	therefore	ADV
ejpam-6813	390	9	:	:	PUNCT
ejpam-6813	390	10	w(2)(f(a	w(2)(f(a	ADJ
ejpam-6813	390	11	)	)	PUNCT
ejpam-6813	390	12	)	)	PUNCT
ejpam-6813	390	13	=	=	PUNCT
ejpam-6813	391	1	diag(w(f(a11	diag(w(f(a11	NOUN
ejpam-6813	391	2	)	)	PUNCT
ejpam-6813	391	3	)	)	PUNCT
ejpam-6813	391	4	,	,	PUNCT
ejpam-6813	391	5	.	.	PUNCT
ejpam-6813	391	6	.	.	PUNCT
ejpam-6813	391	7	.	.	PUNCT
ejpam-6813	392	1	,	,	PUNCT
ejpam-6813	392	2	w(f(amm	w(f(amm	PROPN
ejpam-6813	392	3	)	)	PUNCT
ejpam-6813	392	4	)	)	PUNCT
ejpam-6813	392	5	)	)	PUNCT
ejpam-6813	393	1	≤	≤	NUM
ejpam-6813	393	2	f(w(2)(a	f(w(2)(a	NOUN
ejpam-6813	393	3	)	)	PUNCT
ejpam-6813	393	4	)	)	PUNCT
ejpam-6813	394	1	r.	r.	PROPN
ejpam-6813	394	2	al	al	PROPN
ejpam-6813	394	3	-	-	PROPN
ejpam-6813	394	4	naimi	naimi	PROPN
ejpam-6813	394	5	et	et	PROPN
ejpam-6813	394	6	al	al	PROPN
ejpam-6813	394	7	.	.	PUNCT
ejpam-6813	394	8	/	/	SYM
ejpam-6813	394	9	eur	eur	PROPN
ejpam-6813	394	10	.	.	PUNCT
ejpam-6813	395	1	j.	j.	PROPN
ejpam-6813	395	2	pure	pure	PROPN
ejpam-6813	395	3	appl	appl	PROPN
ejpam-6813	395	4	.	.	PROPN
ejpam-6813	395	5	math	math	PROPN
ejpam-6813	395	6	,	,	PUNCT
ejpam-6813	395	7	18	18	NUM
ejpam-6813	395	8	(	(	PUNCT
ejpam-6813	395	9	4	4	NUM
ejpam-6813	395	10	)	)	PUNCT
ejpam-6813	395	11	(	(	PUNCT
ejpam-6813	395	12	2025	2025	NUM
ejpam-6813	395	13	)	)	PUNCT
ejpam-6813	395	14	,	,	PUNCT
ejpam-6813	395	15	6813	6813	NUM
ejpam-6813	395	16	13	13	NUM
ejpam-6813	395	17	of	of	ADP
ejpam-6813	395	18	15	15	NUM
ejpam-6813	395	19	example	example	NOUN
ejpam-6813	395	20	3	3	NUM
ejpam-6813	395	21	(	(	PUNCT
ejpam-6813	395	22	failure	failure	NOUN
ejpam-6813	395	23	of	of	ADP
ejpam-6813	395	24	triangle	triangle	NOUN
ejpam-6813	395	25	inequality	inequality	NOUN
ejpam-6813	395	26	)	)	PUNCT
ejpam-6813	395	27	.	.	PUNCT
ejpam-6813	396	1	the	the	DET
ejpam-6813	396	2	mapping	mapping	NOUN
ejpam-6813	396	3	a	a	DET
ejpam-6813	396	4	7→	7→	NUM
ejpam-6813	396	5	w(2)(a	w(2)(a	NOUN
ejpam-6813	396	6	)	)	PUNCT
ejpam-6813	396	7	does	do	AUX
ejpam-6813	396	8	not	not	PART
ejpam-6813	396	9	satisfy	satisfy	VERB
ejpam-6813	396	10	the	the	DET
ejpam-6813	396	11	triangle	triangle	NOUN
ejpam-6813	396	12	inequality	inequality	NOUN
ejpam-6813	396	13	.	.	PUNCT
ejpam-6813	397	1	consider	consider	VERB
ejpam-6813	397	2	:	:	PUNCT
ejpam-6813	397	3	a	a	PRON
ejpam-6813	397	4	=	=	X
ejpam-6813	397	5	[	[	PUNCT
ejpam-6813	397	6	0	0	NUM
ejpam-6813	397	7	i	i	NOUN
ejpam-6813	397	8	0	0	NUM
ejpam-6813	397	9	0	0	NUM
ejpam-6813	397	10	]	]	PUNCT
ejpam-6813	397	11	,	,	PUNCT
ejpam-6813	397	12	b	b	X
ejpam-6813	397	13	=	=	PUNCT
ejpam-6813	398	1	[	[	PUNCT
ejpam-6813	398	2	0	0	NUM
ejpam-6813	398	3	0	0	NUM
ejpam-6813	399	1	i	i	NOUN
ejpam-6813	399	2	0	0	NUM
ejpam-6813	399	3	]	]	PUNCT
ejpam-6813	399	4	then	then	ADV
ejpam-6813	399	5	w(2)(a	w(2)(a	NOUN
ejpam-6813	399	6	)	)	PUNCT
ejpam-6813	399	7	=	=	NOUN
ejpam-6813	400	1	[	[	PUNCT
ejpam-6813	400	2	0	0	NUM
ejpam-6813	400	3	1	1	NUM
ejpam-6813	400	4	0	0	NUM
ejpam-6813	400	5	0	0	NUM
ejpam-6813	400	6	]	]	PUNCT
ejpam-6813	400	7	,	,	PUNCT
ejpam-6813	400	8	w(2)(b	w(2)(b	NUM
ejpam-6813	400	9	)	)	PUNCT
ejpam-6813	400	10	=	=	PUNCT
ejpam-6813	401	1	[	[	PUNCT
ejpam-6813	401	2	0	0	NUM
ejpam-6813	401	3	0	0	NUM
ejpam-6813	401	4	1	1	NUM
ejpam-6813	401	5	0	0	NUM
ejpam-6813	401	6	]	]	PUNCT
ejpam-6813	401	7	,	,	PUNCT
ejpam-6813	401	8	and	and	CCONJ
ejpam-6813	401	9	w(2)(a+b	w(2)(a+b	NOUN
ejpam-6813	401	10	)	)	PUNCT
ejpam-6813	401	11	=	=	NOUN
ejpam-6813	402	1	[	[	PUNCT
ejpam-6813	402	2	0	0	NUM
ejpam-6813	402	3	1	1	NUM
ejpam-6813	402	4	1	1	NUM
ejpam-6813	402	5	0	0	NUM
ejpam-6813	402	6	]	]	PUNCT
ejpam-6813	402	7	.	.	PUNCT
ejpam-6813	403	1	we	we	PRON
ejpam-6813	403	2	have	have	VERB
ejpam-6813	403	3	‖w(2)(a+b)‖	‖w(2)(a+b)‖	PROPN
ejpam-6813	403	4	=	=	SYM
ejpam-6813	403	5	1	1	NUM
ejpam-6813	403	6	while	while	SCONJ
ejpam-6813	403	7	‖w(2)(a)‖+	‖w(2)(a)‖+	PROPN
ejpam-6813	403	8	‖w(2)(b)‖	‖w(2)(b)‖	PROPN
ejpam-6813	403	9	=	=	PROPN
ejpam-6813	403	10	1	1	NUM
ejpam-6813	403	11	+	+	SYM
ejpam-6813	403	12	1	1	NUM
ejpam-6813	403	13	=	=	SYM
ejpam-6813	403	14	2	2	NUM
ejpam-6813	403	15	,	,	PUNCT
ejpam-6813	403	16	but	but	CCONJ
ejpam-6813	403	17	the	the	DET
ejpam-6813	403	18	matrix	matrix	NOUN
ejpam-6813	403	19	inequality	inequality	NOUN
ejpam-6813	403	20	w(2)(a	w(2)(a	NOUN
ejpam-6813	404	1	+	+	CCONJ
ejpam-6813	404	2	b	b	X
ejpam-6813	404	3	)	)	PUNCT
ejpam-6813	404	4	6≤	6≤	NUM
ejpam-6813	404	5	w(2)(a	w(2)(a	NOUN
ejpam-6813	404	6	)	)	PUNCT
ejpam-6813	405	1	+	+	CCONJ
ejpam-6813	405	2	w(2)(b	w(2)(b	CCONJ
ejpam-6813	405	3	)	)	PUNCT
ejpam-6813	406	1	demonstrates	demonstrate	VERB
ejpam-6813	406	2	the	the	DET
ejpam-6813	406	3	failure	failure	NOUN
ejpam-6813	406	4	of	of	ADP
ejpam-6813	406	5	the	the	DET
ejpam-6813	406	6	triangle	triangle	NOUN
ejpam-6813	406	7	inequality	inequality	NOUN
ejpam-6813	406	8	at	at	ADP
ejpam-6813	406	9	the	the	DET
ejpam-6813	406	10	matrix	matrix	NOUN
ejpam-6813	406	11	level	level	NOUN
ejpam-6813	406	12	.	.	PUNCT
ejpam-6813	407	1	9	9	X
ejpam-6813	407	2	.	.	X
ejpam-6813	407	3	conclusions	conclusion	NOUN
ejpam-6813	407	4	in	in	ADP
ejpam-6813	407	5	this	this	DET
ejpam-6813	407	6	paper	paper	NOUN
ejpam-6813	407	7	,	,	PUNCT
ejpam-6813	407	8	we	we	PRON
ejpam-6813	407	9	have	have	AUX
ejpam-6813	407	10	successfully	successfully	ADV
ejpam-6813	407	11	established	establish	VERB
ejpam-6813	407	12	a	a	DET
ejpam-6813	407	13	comprehensive	comprehensive	ADJ
ejpam-6813	407	14	theory	theory	NOUN
ejpam-6813	407	15	of	of	ADP
ejpam-6813	407	16	generalized	generalized	ADJ
ejpam-6813	407	17	numerical	numerical	ADJ
ejpam-6813	407	18	radius	radius	NOUN
ejpam-6813	407	19	for	for	ADP
ejpam-6813	407	20	block	block	NOUN
ejpam-6813	407	21	matrix	matrix	NOUN
ejpam-6813	407	22	structures	structure	NOUN
ejpam-6813	407	23	.	.	PUNCT
ejpam-6813	408	1	our	our	PRON
ejpam-6813	408	2	main	main	ADJ
ejpam-6813	408	3	contributions	contribution	NOUN
ejpam-6813	408	4	include	include	VERB
ejpam-6813	408	5	:	:	PUNCT
ejpam-6813	408	6	1	1	X
ejpam-6813	408	7	.	.	X
ejpam-6813	409	1	the	the	DET
ejpam-6813	409	2	introduction	introduction	NOUN
ejpam-6813	409	3	of	of	ADP
ejpam-6813	409	4	partial	partial	ADJ
ejpam-6813	409	5	matrices	matrix	NOUN
ejpam-6813	409	6	of	of	ADP
ejpam-6813	409	7	numerical	numerical	ADJ
ejpam-6813	409	8	radius	radius	PROPN
ejpam-6813	409	9	w(1)(a	w(1)(a	NOUN
ejpam-6813	409	10	)	)	PUNCT
ejpam-6813	409	11	and	and	CCONJ
ejpam-6813	409	12	w(2)(a	w(2)(a	NOUN
ejpam-6813	409	13	)	)	PUNCT
ejpam-6813	409	14	,	,	PUNCT
ejpam-6813	409	15	which	which	PRON
ejpam-6813	409	16	provide	provide	VERB
ejpam-6813	409	17	new	new	ADJ
ejpam-6813	409	18	geometric	geometric	ADJ
ejpam-6813	409	19	insights	insight	NOUN
ejpam-6813	409	20	into	into	ADP
ejpam-6813	409	21	block	block	NOUN
ejpam-6813	409	22	matrix	matrix	NOUN
ejpam-6813	409	23	behavior	behavior	NOUN
ejpam-6813	409	24	.	.	PUNCT
ejpam-6813	410	1	2	2	X
ejpam-6813	410	2	.	.	X
ejpam-6813	410	3	the	the	DET
ejpam-6813	410	4	proof	proof	NOUN
ejpam-6813	410	5	that	that	SCONJ
ejpam-6813	410	6	positive	positive	ADJ
ejpam-6813	410	7	semidefiniteness	semidefiniteness	NOUN
ejpam-6813	410	8	is	be	AUX
ejpam-6813	410	9	preserved	preserve	VERB
ejpam-6813	410	10	under	under	ADP
ejpam-6813	410	11	partial	partial	ADJ
ejpam-6813	410	12	numerical	numerical	ADJ
ejpam-6813	410	13	radius	radius	NOUN
ejpam-6813	410	14	transformations	transformation	NOUN
ejpam-6813	410	15	for	for	ADP
ejpam-6813	410	16	2×	2×	NUM
ejpam-6813	410	17	2	2	NUM
ejpam-6813	410	18	block	block	NOUN
ejpam-6813	410	19	matrices	matrix	NOUN
ejpam-6813	410	20	,	,	PUNCT
ejpam-6813	410	21	with	with	ADP
ejpam-6813	410	22	explicit	explicit	ADJ
ejpam-6813	410	23	characterization	characterization	NOUN
ejpam-6813	410	24	conditions	condition	NOUN
ejpam-6813	410	25	.	.	PUNCT
ejpam-6813	411	1	3	3	X
ejpam-6813	411	2	.	.	X
ejpam-6813	411	3	the	the	DET
ejpam-6813	411	4	establishment	establishment	NOUN
ejpam-6813	411	5	of	of	ADP
ejpam-6813	411	6	fundamental	fundamental	ADJ
ejpam-6813	411	7	properties	property	NOUN
ejpam-6813	411	8	including	include	VERB
ejpam-6813	411	9	homogeneity	homogeneity	NOUN
ejpam-6813	411	10	,	,	PUNCT
ejpam-6813	411	11	unitary	unitary	ADJ
ejpam-6813	411	12	invariance	invariance	NOUN
ejpam-6813	411	13	,	,	PUNCT
ejpam-6813	411	14	and	and	CCONJ
ejpam-6813	411	15	structural	structural	ADJ
ejpam-6813	411	16	relationships	relationship	NOUN
ejpam-6813	411	17	for	for	ADP
ejpam-6813	411	18	various	various	ADJ
ejpam-6813	411	19	classes	class	NOUN
ejpam-6813	411	20	of	of	ADP
ejpam-6813	411	21	block	block	NOUN
ejpam-6813	411	22	matrices	matrix	NOUN
ejpam-6813	411	23	.	.	PUNCT
ejpam-6813	412	1	4	4	X
ejpam-6813	412	2	.	.	X
ejpam-6813	412	3	the	the	DET
ejpam-6813	412	4	development	development	NOUN
ejpam-6813	412	5	of	of	ADP
ejpam-6813	412	6	bounds	bound	NOUN
ejpam-6813	412	7	and	and	CCONJ
ejpam-6813	412	8	inequalities	inequality	NOUN
ejpam-6813	412	9	connecting	connect	VERB
ejpam-6813	412	10	our	our	PRON
ejpam-6813	412	11	new	new	ADJ
ejpam-6813	412	12	constructions	construction	NOUN
ejpam-6813	412	13	to	to	ADP
ejpam-6813	412	14	classical	classical	ADJ
ejpam-6813	412	15	matrix	matrix	NOUN
ejpam-6813	412	16	parameters	parameter	NOUN
ejpam-6813	412	17	.	.	PUNCT
ejpam-6813	413	1	these	these	DET
ejpam-6813	413	2	results	result	NOUN
ejpam-6813	413	3	extend	extend	VERB
ejpam-6813	413	4	classical	classical	ADJ
ejpam-6813	413	5	numerical	numerical	ADJ
ejpam-6813	413	6	radius	radius	PROPN
ejpam-6813	413	7	theory	theory	NOUN
ejpam-6813	413	8	to	to	PART
ejpam-6813	413	9	block	block	NOUN
ejpam-6813	413	10	matrix	matrix	NOUN
ejpam-6813	413	11	frameworks	framework	NOUN
ejpam-6813	413	12	while	while	SCONJ
ejpam-6813	413	13	preserving	preserve	VERB
ejpam-6813	413	14	essential	essential	ADJ
ejpam-6813	413	15	geometric	geometric	ADJ
ejpam-6813	413	16	and	and	CCONJ
ejpam-6813	413	17	analytical	analytical	ADJ
ejpam-6813	413	18	properties	property	NOUN
ejpam-6813	413	19	.	.	PUNCT
ejpam-6813	414	1	the	the	DET
ejpam-6813	414	2	positive	positive	ADJ
ejpam-6813	414	3	semidefiniteness	semidefiniteness	ADJ
ejpam-6813	414	4	preservation	preservation	NOUN
ejpam-6813	414	5	result	result	NOUN
ejpam-6813	414	6	for	for	ADP
ejpam-6813	414	7	2	2	NUM
ejpam-6813	414	8	×	×	NOUN
ejpam-6813	414	9	2	2	NUM
ejpam-6813	414	10	blocks	block	NOUN
ejpam-6813	414	11	is	be	AUX
ejpam-6813	414	12	particularly	particularly	ADV
ejpam-6813	414	13	significant	significant	ADJ
ejpam-6813	414	14	,	,	PUNCT
ejpam-6813	414	15	providing	provide	VERB
ejpam-6813	414	16	both	both	CCONJ
ejpam-6813	414	17	theoretical	theoretical	ADJ
ejpam-6813	414	18	understanding	understanding	NOUN
ejpam-6813	414	19	and	and	CCONJ
ejpam-6813	414	20	practical	practical	ADJ
ejpam-6813	414	21	computational	computational	ADJ
ejpam-6813	414	22	advantages	advantage	NOUN
ejpam-6813	414	23	.	.	PUNCT
ejpam-6813	415	1	our	our	PRON
ejpam-6813	415	2	work	work	NOUN
ejpam-6813	415	3	opens	open	VERB
ejpam-6813	415	4	several	several	ADJ
ejpam-6813	415	5	avenues	avenue	NOUN
ejpam-6813	415	6	for	for	ADP
ejpam-6813	415	7	future	future	ADJ
ejpam-6813	415	8	research	research	NOUN
ejpam-6813	415	9	.	.	PUNCT
ejpam-6813	416	1	natural	natural	ADJ
ejpam-6813	416	2	extensions	extension	NOUN
ejpam-6813	416	3	include	include	VERB
ejpam-6813	416	4	infinite	infinite	ADJ
ejpam-6813	416	5	-	-	PUNCT
ejpam-6813	416	6	dimensional	dimensional	ADJ
ejpam-6813	416	7	block	block	NOUN
ejpam-6813	416	8	operators	operator	NOUN
ejpam-6813	416	9	,	,	PUNCT
ejpam-6813	416	10	where	where	SCONJ
ejpam-6813	416	11	connections	connection	NOUN
ejpam-6813	416	12	to	to	ADP
ejpam-6813	416	13	the	the	DET
ejpam-6813	416	14	p	p	PROPN
ejpam-6813	416	15	-	-	PUNCT
ejpam-6813	416	16	numerical	numerical	ADJ
ejpam-6813	416	17	radius	radius	PROPN
ejpam-6813	416	18	theory	theory	NOUN
ejpam-6813	416	19	[	[	X
ejpam-6813	416	20	15	15	NUM
ejpam-6813	416	21	]	]	PUNCT
ejpam-6813	416	22	may	may	AUX
ejpam-6813	416	23	yield	yield	VERB
ejpam-6813	416	24	fruitful	fruitful	ADJ
ejpam-6813	416	25	insights	insight	NOUN
ejpam-6813	416	26	.	.	PUNCT
ejpam-6813	417	1	the	the	DET
ejpam-6813	417	2	relationship	relationship	NOUN
ejpam-6813	417	3	between	between	ADP
ejpam-6813	417	4	our	our	PRON
ejpam-6813	417	5	partial	partial	ADJ
ejpam-6813	417	6	numerical	numerical	ADJ
ejpam-6813	417	7	radius	radius	NOUN
ejpam-6813	417	8	framework	framework	NOUN
ejpam-6813	417	9	and	and	CCONJ
ejpam-6813	417	10	the	the	DET
ejpam-6813	417	11	partial	partial	ADJ
ejpam-6813	417	12	spectral	spectral	ADJ
ejpam-6813	417	13	radius	radius	NOUN
ejpam-6813	417	14	approach	approach	NOUN
ejpam-6813	417	15	developed	develop	VERB
ejpam-6813	417	16	in	in	ADP
ejpam-6813	417	17	[	[	X
ejpam-6813	417	18	14	14	NUM
ejpam-6813	417	19	]	]	PUNCT
ejpam-6813	417	20	suggests	suggest	VERB
ejpam-6813	417	21	opportunities	opportunity	NOUN
ejpam-6813	417	22	for	for	ADP
ejpam-6813	417	23	unified	unified	ADJ
ejpam-6813	417	24	treatments	treatment	NOUN
ejpam-6813	417	25	combining	combine	VERB
ejpam-6813	417	26	spectral	spectral	ADJ
ejpam-6813	417	27	and	and	CCONJ
ejpam-6813	417	28	geometric	geometric	ADJ
ejpam-6813	417	29	perspectives	perspective	NOUN
ejpam-6813	417	30	.	.	PUNCT
ejpam-6813	418	1	additional	additional	ADJ
ejpam-6813	418	2	directions	direction	NOUN
ejpam-6813	418	3	include	include	VERB
ejpam-6813	418	4	applications	application	NOUN
ejpam-6813	418	5	to	to	PART
ejpam-6813	418	6	matrix	matrix	VERB
ejpam-6813	418	7	completion	completion	NOUN
ejpam-6813	418	8	problems	problem	NOUN
ejpam-6813	418	9	,	,	PUNCT
ejpam-6813	418	10	connections	connection	NOUN
ejpam-6813	418	11	to	to	ADP
ejpam-6813	418	12	quantum	quantum	ADJ
ejpam-6813	418	13	information	information	NOUN
ejpam-6813	418	14	theory	theory	NOUN
ejpam-6813	418	15	,	,	PUNCT
ejpam-6813	418	16	and	and	CCONJ
ejpam-6813	418	17	extensions	extension	NOUN
ejpam-6813	418	18	to	to	ADP
ejpam-6813	418	19	more	more	ADV
ejpam-6813	418	20	general	general	ADJ
ejpam-6813	418	21	unitarily	unitarily	ADV
ejpam-6813	418	22	invariant	invariant	ADJ
ejpam-6813	418	23	norms	norm	NOUN
ejpam-6813	418	24	beyond	beyond	ADP
ejpam-6813	418	25	those	those	PRON
ejpam-6813	418	26	considered	consider	VERB
ejpam-6813	418	27	here	here	ADV
ejpam-6813	418	28	.	.	PUNCT
ejpam-6813	419	1	the	the	DET
ejpam-6813	419	2	interplay	interplay	NOUN
ejpam-6813	419	3	between	between	ADP
ejpam-6813	419	4	block	block	NOUN
ejpam-6813	419	5	structure	structure	NOUN
ejpam-6813	419	6	and	and	CCONJ
ejpam-6813	419	7	numerical	numerical	ADJ
ejpam-6813	419	8	radius	radius	NOUN
ejpam-6813	419	9	properties	property	NOUN
ejpam-6813	419	10	suggests	suggest	VERB
ejpam-6813	419	11	rich	rich	ADJ
ejpam-6813	419	12	possibilities	possibility	NOUN
ejpam-6813	419	13	for	for	ADP
ejpam-6813	419	14	further	further	ADJ
ejpam-6813	419	15	exploration	exploration	NOUN
ejpam-6813	419	16	in	in	ADP
ejpam-6813	419	17	operator	operator	NOUN
ejpam-6813	419	18	theory	theory	NOUN
ejpam-6813	419	19	and	and	CCONJ
ejpam-6813	419	20	matrix	matrix	NOUN
ejpam-6813	419	21	analysis	analysis	NOUN
ejpam-6813	419	22	.	.	PUNCT
ejpam-6813	420	1	the	the	DET
ejpam-6813	420	2	limitations	limitation	NOUN
ejpam-6813	420	3	we	we	PRON
ejpam-6813	420	4	have	have	AUX
ejpam-6813	420	5	identified	identify	VERB
ejpam-6813	420	6	,	,	PUNCT
ejpam-6813	420	7	such	such	ADJ
ejpam-6813	420	8	as	as	ADP
ejpam-6813	420	9	the	the	DET
ejpam-6813	420	10	failure	failure	NOUN
ejpam-6813	420	11	of	of	ADP
ejpam-6813	420	12	the	the	DET
ejpam-6813	420	13	triangle	triangle	NOUN
ejpam-6813	420	14	inequality	inequality	NOUN
ejpam-6813	420	15	and	and	CCONJ
ejpam-6813	420	16	the	the	DET
ejpam-6813	420	17	restriction	restriction	NOUN
ejpam-6813	420	18	of	of	ADP
ejpam-6813	420	19	positive	positive	ADJ
ejpam-6813	420	20	semidefiniteness	semidefiniteness	ADJ
ejpam-6813	420	21	preservation	preservation	NOUN
ejpam-6813	420	22	to	to	ADP
ejpam-6813	420	23	2	2	NUM
ejpam-6813	420	24	×	×	NOUN
ejpam-6813	420	25	2	2	NUM
ejpam-6813	420	26	blocks	block	NOUN
ejpam-6813	420	27	,	,	PUNCT
ejpam-6813	420	28	provide	provide	VERB
ejpam-6813	420	29	r.	r.	PROPN
ejpam-6813	420	30	al	al	PROPN
ejpam-6813	420	31	-	-	PROPN
ejpam-6813	420	32	naimi	naimi	PROPN
ejpam-6813	420	33	et	et	PROPN
ejpam-6813	420	34	al	al	PROPN
ejpam-6813	420	35	.	.	PUNCT
ejpam-6813	420	36	/	/	SYM
ejpam-6813	420	37	eur	eur	PROPN
ejpam-6813	420	38	.	.	PUNCT
ejpam-6813	421	1	j.	j.	PROPN
ejpam-6813	421	2	pure	pure	PROPN
ejpam-6813	421	3	appl	appl	PROPN
ejpam-6813	421	4	.	.	PROPN
ejpam-6813	421	5	math	math	PROPN
ejpam-6813	421	6	,	,	PUNCT
ejpam-6813	421	7	18	18	NUM
ejpam-6813	421	8	(	(	PUNCT
ejpam-6813	421	9	4	4	NUM
ejpam-6813	421	10	)	)	PUNCT
ejpam-6813	421	11	(	(	PUNCT
ejpam-6813	421	12	2025	2025	NUM
ejpam-6813	421	13	)	)	PUNCT
ejpam-6813	421	14	,	,	PUNCT
ejpam-6813	421	15	6813	6813	NUM
ejpam-6813	421	16	14	14	NUM
ejpam-6813	421	17	of	of	ADP
ejpam-6813	421	18	15	15	NUM
ejpam-6813	421	19	important	important	ADJ
ejpam-6813	421	20	boundaries	boundary	NOUN
ejpam-6813	421	21	for	for	ADP
ejpam-6813	421	22	the	the	DET
ejpam-6813	421	23	theory	theory	NOUN
ejpam-6813	421	24	and	and	CCONJ
ejpam-6813	421	25	suggest	suggest	VERB
ejpam-6813	421	26	directions	direction	NOUN
ejpam-6813	421	27	for	for	ADP
ejpam-6813	421	28	refined	refined	ADJ
ejpam-6813	421	29	approaches	approach	NOUN
ejpam-6813	421	30	in	in	ADP
ejpam-6813	421	31	future	future	ADJ
ejpam-6813	421	32	investigations	investigation	NOUN
ejpam-6813	421	33	.	.	PUNCT
ejpam-6813	422	1	data	datum	NOUN
ejpam-6813	422	2	availability	availability	NOUN
ejpam-6813	422	3	statement	statement	NOUN
ejpam-6813	422	4	funding	funding	NOUN
ejpam-6813	422	5	:	:	PUNCT
ejpam-6813	422	6	not	not	PART
ejpam-6813	422	7	applicable	applicable	ADJ
ejpam-6813	422	8	conflicts	conflict	NOUN
ejpam-6813	422	9	of	of	ADP
ejpam-6813	422	10	interest	interest	NOUN
ejpam-6813	422	11	/	/	SYM
ejpam-6813	422	12	competing	compete	VERB
ejpam-6813	422	13	interests	interest	NOUN
ejpam-6813	422	14	:	:	PUNCT
ejpam-6813	422	15	not	not	PART
ejpam-6813	422	16	applicable	applicable	ADJ
ejpam-6813	422	17	availability	availability	NOUN
ejpam-6813	422	18	of	of	ADP
ejpam-6813	422	19	data	datum	NOUN
ejpam-6813	422	20	and	and	CCONJ
ejpam-6813	422	21	material	material	NOUN
ejpam-6813	422	22	:	:	PUNCT
ejpam-6813	422	23	not	not	PART
ejpam-6813	422	24	applicable	applicable	ADJ
ejpam-6813	422	25	code	code	NOUN
ejpam-6813	422	26	availability	availability	NOUN
ejpam-6813	422	27	:	:	PUNCT
ejpam-6813	422	28	not	not	PART
ejpam-6813	422	29	applicable	applicable	ADJ
ejpam-6813	422	30	author	author	NOUN
ejpam-6813	422	31	’s	’s	PART
ejpam-6813	422	32	contributions	contribution	NOUN
ejpam-6813	422	33	:	:	PUNCT
ejpam-6813	422	34	all	all	DET
ejpam-6813	422	35	authors	author	NOUN
ejpam-6813	422	36	contributed	contribute	VERB
ejpam-6813	422	37	equally	equally	ADV
ejpam-6813	422	38	to	to	ADP
ejpam-6813	422	39	all	all	DET
ejpam-6813	422	40	parts	part	NOUN
ejpam-6813	422	41	of	of	ADP
ejpam-6813	422	42	the	the	DET
ejpam-6813	422	43	paper	paper	NOUN
ejpam-6813	422	44	.	.	PUNCT
ejpam-6813	423	1	references	reference	NOUN
ejpam-6813	423	2	[	[	X
ejpam-6813	423	3	1	1	NUM
ejpam-6813	423	4	]	]	PUNCT
ejpam-6813	423	5	r.	r.	PROPN
ejpam-6813	423	6	bhatia	bhatia	PROPN
ejpam-6813	423	7	.	.	PUNCT
ejpam-6813	424	1	matrix	matrix	NOUN
ejpam-6813	424	2	analysis	analysis	NOUN
ejpam-6813	424	3	.	.	PUNCT
ejpam-6813	425	1	springer	springer	NOUN
ejpam-6813	425	2	,	,	PUNCT
ejpam-6813	425	3	new	new	PROPN
ejpam-6813	425	4	york	york	PROPN
ejpam-6813	425	5	,	,	PUNCT
ejpam-6813	425	6	1997	1997	NUM
ejpam-6813	425	7	.	.	PUNCT
ejpam-6813	426	1	[	[	X
ejpam-6813	426	2	2	2	NUM
ejpam-6813	426	3	]	]	PUNCT
ejpam-6813	426	4	r.	r.	PROPN
ejpam-6813	426	5	a.	a.	NOUN
ejpam-6813	426	6	horn	horn	PROPN
ejpam-6813	426	7	and	and	CCONJ
ejpam-6813	426	8	c.	c.	PROPN
ejpam-6813	426	9	r.	r.	PROPN
ejpam-6813	426	10	johnson	johnson	PROPN
ejpam-6813	426	11	.	.	PUNCT
ejpam-6813	426	12	matrix	matrix	NOUN
ejpam-6813	426	13	analysis	analysis	NOUN
ejpam-6813	426	14	.	.	PUNCT
ejpam-6813	427	1	cambridge	cambridge	PROPN
ejpam-6813	427	2	university	university	PROPN
ejpam-6813	427	3	press	press	PROPN
ejpam-6813	427	4	,	,	PUNCT
ejpam-6813	427	5	cambridge	cambridge	PROPN
ejpam-6813	427	6	,	,	PUNCT
ejpam-6813	427	7	2	2	NUM
ejpam-6813	427	8	edition	edition	NOUN
ejpam-6813	427	9	,	,	PUNCT
ejpam-6813	427	10	2013	2013	NUM
ejpam-6813	427	11	.	.	PUNCT
ejpam-6813	428	1	[	[	X
ejpam-6813	428	2	3	3	X
ejpam-6813	428	3	]	]	X
ejpam-6813	428	4	f.	f.	PROPN
ejpam-6813	428	5	kittaneh	kittaneh	PROPN
ejpam-6813	428	6	.	.	PUNCT
ejpam-6813	429	1	spectral	spectral	ADJ
ejpam-6813	429	2	radius	radius	NOUN
ejpam-6813	429	3	inequalities	inequality	NOUN
ejpam-6813	429	4	for	for	ADP
ejpam-6813	429	5	hilbert	hilbert	NOUN
ejpam-6813	429	6	space	space	NOUN
ejpam-6813	429	7	operators	operator	NOUN
ejpam-6813	429	8	.	.	PUNCT
ejpam-6813	430	1	proceedings	proceeding	NOUN
ejpam-6813	430	2	of	of	ADP
ejpam-6813	430	3	the	the	DET
ejpam-6813	430	4	american	american	PROPN
ejpam-6813	430	5	mathematical	mathematical	PROPN
ejpam-6813	430	6	society	society	NOUN
ejpam-6813	430	7	,	,	PUNCT
ejpam-6813	430	8	134:385–390	134:385–390	NUM
ejpam-6813	430	9	,	,	PUNCT
ejpam-6813	430	10	2006	2006	NUM
ejpam-6813	430	11	.	.	PUNCT
ejpam-6813	431	1	[	[	X
ejpam-6813	431	2	4	4	X
ejpam-6813	431	3	]	]	PUNCT
ejpam-6813	431	4	w.	w.	NOUN
ejpam-6813	431	5	audeh	audeh	PROPN
ejpam-6813	431	6	.	.	PUNCT
ejpam-6813	432	1	some	some	DET
ejpam-6813	432	2	generalizations	generalization	NOUN
ejpam-6813	432	3	for	for	ADP
ejpam-6813	432	4	singular	singular	ADJ
ejpam-6813	432	5	value	value	NOUN
ejpam-6813	432	6	inequalities	inequality	NOUN
ejpam-6813	432	7	of	of	ADP
ejpam-6813	432	8	compact	compact	ADJ
ejpam-6813	432	9	operators	operator	NOUN
ejpam-6813	432	10	.	.	PUNCT
ejpam-6813	433	1	advances	advance	NOUN
ejpam-6813	433	2	in	in	ADP
ejpam-6813	433	3	operator	operator	NOUN
ejpam-6813	433	4	theory	theory	NOUN
ejpam-6813	433	5	,	,	PUNCT
ejpam-6813	433	6	6:1–10	6:1–10	NOUN
ejpam-6813	433	7	,	,	PUNCT
ejpam-6813	433	8	2021	2021	NUM
ejpam-6813	433	9	.	.	PUNCT
ejpam-6813	434	1	[	[	X
ejpam-6813	434	2	5	5	X
ejpam-6813	434	3	]	]	PUNCT
ejpam-6813	434	4	w.	w.	NOUN
ejpam-6813	434	5	audeh	audeh	PROPN
ejpam-6813	434	6	.	.	PUNCT
ejpam-6813	435	1	generalizations	generalization	NOUN
ejpam-6813	435	2	for	for	ADP
ejpam-6813	435	3	singular	singular	ADJ
ejpam-6813	435	4	value	value	NOUN
ejpam-6813	435	5	and	and	CCONJ
ejpam-6813	435	6	arithmetic	arithmetic	ADJ
ejpam-6813	435	7	-	-	PUNCT
ejpam-6813	435	8	geometric	geometric	ADJ
ejpam-6813	435	9	mean	mean	NOUN
ejpam-6813	435	10	inequalities	inequality	NOUN
ejpam-6813	435	11	of	of	ADP
ejpam-6813	435	12	operators	operator	NOUN
ejpam-6813	435	13	.	.	PUNCT
ejpam-6813	436	1	journal	journal	PROPN
ejpam-6813	436	2	of	of	ADP
ejpam-6813	436	3	mathematical	mathematical	ADJ
ejpam-6813	436	4	analysis	analysis	NOUN
ejpam-6813	436	5	and	and	CCONJ
ejpam-6813	436	6	applications	application	NOUN
ejpam-6813	436	7	,	,	PUNCT
ejpam-6813	436	8	489:1–8	489:1–8	NUM
ejpam-6813	436	9	,	,	PUNCT
ejpam-6813	436	10	2020	2020	NUM
ejpam-6813	436	11	.	.	PUNCT
ejpam-6813	437	1	[	[	X
ejpam-6813	437	2	6	6	NUM
ejpam-6813	437	3	]	]	PUNCT
ejpam-6813	437	4	w.	w.	NOUN
ejpam-6813	437	5	audeh	audeh	PROPN
ejpam-6813	437	6	.	.	PUNCT
ejpam-6813	438	1	generalizations	generalization	NOUN
ejpam-6813	438	2	for	for	ADP
ejpam-6813	438	3	singular	singular	ADJ
ejpam-6813	438	4	value	value	NOUN
ejpam-6813	438	5	inequalities	inequality	NOUN
ejpam-6813	438	6	of	of	ADP
ejpam-6813	438	7	operators	operator	NOUN
ejpam-6813	438	8	.	.	PUNCT
ejpam-6813	439	1	advances	advance	NOUN
ejpam-6813	439	2	in	in	ADP
ejpam-6813	439	3	operator	operator	NOUN
ejpam-6813	439	4	theory	theory	NOUN
ejpam-6813	439	5	,	,	PUNCT
ejpam-6813	439	6	5:371–381	5:371–381	PROPN
ejpam-6813	439	7	,	,	PUNCT
ejpam-6813	439	8	2020	2020	NUM
ejpam-6813	439	9	.	.	PUNCT
ejpam-6813	440	1	[	[	X
ejpam-6813	440	2	7	7	X
ejpam-6813	440	3	]	]	PUNCT
ejpam-6813	440	4	w.	w.	NOUN
ejpam-6813	440	5	audeh	audeh	PROPN
ejpam-6813	440	6	.	.	PUNCT
ejpam-6813	441	1	singular	singular	PROPN
ejpam-6813	441	2	value	value	NOUN
ejpam-6813	441	3	inequalities	inequality	NOUN
ejpam-6813	441	4	and	and	CCONJ
ejpam-6813	441	5	applications	application	NOUN
ejpam-6813	441	6	.	.	PUNCT
ejpam-6813	442	1	positivity	positivity	NOUN
ejpam-6813	442	2	,	,	PUNCT
ejpam-6813	442	3	25:843–852	25:843–852	PROPN
ejpam-6813	442	4	,	,	PUNCT
ejpam-6813	442	5	2020	2020	NUM
ejpam-6813	442	6	.	.	PUNCT
ejpam-6813	443	1	[	[	X
ejpam-6813	443	2	8	8	X
ejpam-6813	443	3	]	]	X
ejpam-6813	443	4	w.	w.	NOUN
ejpam-6813	443	5	audeh	audeh	PROPN
ejpam-6813	443	6	and	and	CCONJ
ejpam-6813	443	7	f.	f.	PROPN
ejpam-6813	443	8	kittaneh	kittaneh	PROPN
ejpam-6813	443	9	.	.	PUNCT
ejpam-6813	444	1	singular	singular	PROPN
ejpam-6813	444	2	value	value	NOUN
ejpam-6813	444	3	inequalities	inequality	NOUN
ejpam-6813	444	4	for	for	ADP
ejpam-6813	444	5	compact	compact	ADJ
ejpam-6813	444	6	operators	operator	NOUN
ejpam-6813	444	7	.	.	PUNCT
ejpam-6813	445	1	linear	linear	ADJ
ejpam-6813	445	2	algebra	algebra	NOUN
ejpam-6813	445	3	and	and	CCONJ
ejpam-6813	445	4	its	its	PRON
ejpam-6813	445	5	applications	application	NOUN
ejpam-6813	445	6	,	,	PUNCT
ejpam-6813	445	7	437:2516–2522	437:2516–2522	NOUN
ejpam-6813	445	8	,	,	PUNCT
ejpam-6813	445	9	2012	2012	NUM
ejpam-6813	445	10	.	.	PUNCT
ejpam-6813	446	1	[	[	X
ejpam-6813	446	2	9	9	NUM
ejpam-6813	446	3	]	]	X
ejpam-6813	446	4	r.	r.	PROPN
ejpam-6813	446	5	a.	a.	PROPN
ejpam-6813	446	6	horn	horn	PROPN
ejpam-6813	446	7	,	,	PUNCT
ejpam-6813	446	8	r.	r.	PROPN
ejpam-6813	446	9	mathias	mathias	PROPN
ejpam-6813	446	10	,	,	PUNCT
ejpam-6813	446	11	and	and	CCONJ
ejpam-6813	446	12	y.	y.	PROPN
ejpam-6813	446	13	nakamura	nakamura	PROPN
ejpam-6813	446	14	.	.	PUNCT
ejpam-6813	447	1	inequalities	inequality	NOUN
ejpam-6813	447	2	for	for	ADP
ejpam-6813	447	3	unitarily	unitarily	ADV
ejpam-6813	447	4	invariant	invariant	ADJ
ejpam-6813	447	5	norms	norm	NOUN
ejpam-6813	447	6	and	and	CCONJ
ejpam-6813	447	7	bilinear	bilinear	NOUN
ejpam-6813	447	8	matrix	matrix	NOUN
ejpam-6813	447	9	products	product	NOUN
ejpam-6813	447	10	.	.	PUNCT
ejpam-6813	448	1	linear	linear	ADJ
ejpam-6813	448	2	and	and	CCONJ
ejpam-6813	448	3	multilinear	multilinear	PROPN
ejpam-6813	448	4	algebra	algebra	PROPN
ejpam-6813	448	5	,	,	PUNCT
ejpam-6813	448	6	30:303–314	30:303–314	PROPN
ejpam-6813	448	7	,	,	PUNCT
ejpam-6813	448	8	1991	1991	NUM
ejpam-6813	448	9	.	.	PUNCT
ejpam-6813	449	1	[	[	X
ejpam-6813	449	2	10	10	NUM
ejpam-6813	449	3	]	]	PUNCT
ejpam-6813	449	4	z.	z.	PROPN
ejpam-6813	449	5	kishka	kishka	PROPN
ejpam-6813	449	6	and	and	CCONJ
ejpam-6813	449	7	m.	m.	PROPN
ejpam-6813	449	8	s.	s.	PROPN
ejpam-6813	449	9	abdalla	abdalla	PROPN
ejpam-6813	449	10	.	.	PUNCT
ejpam-6813	450	1	on	on	ADP
ejpam-6813	450	2	hadamard	hadamard	NOUN
ejpam-6813	450	3	and	and	CCONJ
ejpam-6813	450	4	kronecker	kronecker	NOUN
ejpam-6813	450	5	products	product	NOUN
ejpam-6813	450	6	over	over	ADP
ejpam-6813	450	7	matrix	matrix	NOUN
ejpam-6813	450	8	of	of	ADP
ejpam-6813	450	9	matrices	matrix	NOUN
ejpam-6813	450	10	.	.	PUNCT
ejpam-6813	451	1	general	general	ADJ
ejpam-6813	451	2	letters	letter	NOUN
ejpam-6813	451	3	in	in	ADP
ejpam-6813	451	4	mathematics	mathematic	NOUN
ejpam-6813	451	5	,	,	PUNCT
ejpam-6813	451	6	4:13–22	4:13–22	NUM
ejpam-6813	451	7	,	,	PUNCT
ejpam-6813	451	8	2018	2018	NUM
ejpam-6813	451	9	.	.	PUNCT
ejpam-6813	452	1	[	[	X
ejpam-6813	452	2	11	11	NUM
ejpam-6813	452	3	]	]	X
ejpam-6813	452	4	c.	c.	PROPN
ejpam-6813	452	5	fitzgerald	fitzgerald	PROPN
ejpam-6813	452	6	and	and	CCONJ
ejpam-6813	452	7	r.	r.	PROPN
ejpam-6813	452	8	a.	a.	PROPN
ejpam-6813	452	9	horn	horn	PROPN
ejpam-6813	452	10	.	.	PUNCT
ejpam-6813	453	1	on	on	ADP
ejpam-6813	453	2	fractional	fractional	ADJ
ejpam-6813	453	3	hadamard	hadamard	ADJ
ejpam-6813	453	4	powers	power	NOUN
ejpam-6813	453	5	of	of	ADP
ejpam-6813	453	6	positive	positive	ADJ
ejpam-6813	453	7	definite	definite	ADJ
ejpam-6813	453	8	matrices	matrix	NOUN
ejpam-6813	453	9	.	.	PUNCT
ejpam-6813	454	1	journal	journal	PROPN
ejpam-6813	454	2	of	of	ADP
ejpam-6813	454	3	mathematical	mathematical	ADJ
ejpam-6813	454	4	analysis	analysis	NOUN
ejpam-6813	454	5	and	and	CCONJ
ejpam-6813	454	6	applications	application	NOUN
ejpam-6813	454	7	,	,	PUNCT
ejpam-6813	454	8	61:633–642	61:633–642	NUM
ejpam-6813	454	9	,	,	PUNCT
ejpam-6813	454	10	1977	1977	NUM
ejpam-6813	454	11	.	.	PUNCT
ejpam-6813	455	1	[	[	X
ejpam-6813	455	2	12	12	NUM
ejpam-6813	455	3	]	]	PUNCT
ejpam-6813	455	4	a.	a.	NOUN
ejpam-6813	455	5	abu	abu	PROPN
ejpam-6813	455	6	-	-	PUNCT
ejpam-6813	455	7	omar	omar	PROPN
ejpam-6813	455	8	and	and	CCONJ
ejpam-6813	455	9	f.	f.	PROPN
ejpam-6813	455	10	kittaneh	kittaneh	PROPN
ejpam-6813	455	11	.	.	PUNCT
ejpam-6813	456	1	numerical	numerical	PROPN
ejpam-6813	456	2	radius	radius	PROPN
ejpam-6813	456	3	inequalities	inequality	NOUN
ejpam-6813	456	4	for	for	ADP
ejpam-6813	456	5	n	n	NUM
ejpam-6813	456	6	×	×	NOUN
ejpam-6813	456	7	n	n	PRON
ejpam-6813	456	8	operator	operator	NOUN
ejpam-6813	456	9	matrices	matrix	NOUN
ejpam-6813	456	10	.	.	PUNCT
ejpam-6813	457	1	linear	linear	ADJ
ejpam-6813	457	2	algebra	algebra	NOUN
ejpam-6813	457	3	and	and	CCONJ
ejpam-6813	457	4	its	its	PRON
ejpam-6813	457	5	applications	application	NOUN
ejpam-6813	457	6	,	,	PUNCT
ejpam-6813	457	7	468:18–26	468:18–26	NUM
ejpam-6813	457	8	,	,	PUNCT
ejpam-6813	457	9	2015	2015	NUM
ejpam-6813	457	10	.	.	PUNCT
ejpam-6813	458	1	[	[	X
ejpam-6813	458	2	13	13	NUM
ejpam-6813	458	3	]	]	PUNCT
ejpam-6813	458	4	m.	m.	NOUN
ejpam-6813	458	5	al	al	PROPN
ejpam-6813	458	6	-	-	PUNCT
ejpam-6813	458	7	labadi	labadi	PROPN
ejpam-6813	458	8	,	,	PUNCT
ejpam-6813	458	9	r.	r.	PROPN
ejpam-6813	458	10	al	al	PROPN
ejpam-6813	458	11	-	-	PUNCT
ejpam-6813	458	12	naimi	naimi	PROPN
ejpam-6813	458	13	,	,	PUNCT
ejpam-6813	458	14	and	and	CCONJ
ejpam-6813	458	15	w.	w.	PROPN
ejpam-6813	458	16	audeh	audeh	PROPN
ejpam-6813	458	17	.	.	PUNCT
ejpam-6813	459	1	partial	partial	ADJ
ejpam-6813	459	2	eigenvalues	eigenvalue	VERB
ejpam-6813	459	3	for	for	ADP
ejpam-6813	459	4	block	block	NOUN
ejpam-6813	459	5	matrices	matrix	NOUN
ejpam-6813	459	6	.	.	PUNCT
ejpam-6813	460	1	journal	journal	PROPN
ejpam-6813	460	2	of	of	ADP
ejpam-6813	460	3	mathematical	mathematical	ADJ
ejpam-6813	460	4	inequalities	inequality	NOUN
ejpam-6813	460	5	,	,	PUNCT
ejpam-6813	460	6	18:991–1005	18:991–1005	NUM
ejpam-6813	460	7	,	,	PUNCT
ejpam-6813	460	8	2024	2024	NUM
ejpam-6813	460	9	.	.	PUNCT
ejpam-6813	461	1	[	[	X
ejpam-6813	461	2	14	14	NUM
ejpam-6813	461	3	]	]	PUNCT
ejpam-6813	461	4	r.	r.	PROPN
ejpam-6813	461	5	al	al	PROPN
ejpam-6813	461	6	-	-	PUNCT
ejpam-6813	461	7	naimi	naimi	PROPN
ejpam-6813	461	8	,	,	PUNCT
ejpam-6813	461	9	m.	m.	NOUN
ejpam-6813	461	10	al	al	PROPN
ejpam-6813	461	11	-	-	PUNCT
ejpam-6813	461	12	labadi	labadi	NOUN
ejpam-6813	461	13	,	,	PUNCT
ejpam-6813	461	14	and	and	CCONJ
ejpam-6813	461	15	w.	w.	PROPN
ejpam-6813	461	16	audeh	audeh	PROPN
ejpam-6813	461	17	.	.	PUNCT
ejpam-6813	462	1	partial	partial	ADJ
ejpam-6813	462	2	matrix	matrix	NOUN
ejpam-6813	462	3	of	of	ADP
ejpam-6813	462	4	spectral	spectral	ADJ
ejpam-6813	462	5	radius	radius	NOUN
ejpam-6813	462	6	and	and	CCONJ
ejpam-6813	462	7	partial	partial	ADJ
ejpam-6813	462	8	matrix	matrix	NOUN
ejpam-6813	462	9	of	of	ADP
ejpam-6813	462	10	norm	norm	NOUN
ejpam-6813	462	11	inequalities	inequality	NOUN
ejpam-6813	462	12	for	for	ADP
ejpam-6813	462	13	block	block	NOUN
ejpam-6813	462	14	matrices	matrix	NOUN
ejpam-6813	462	15	.	.	PUNCT
ejpam-6813	463	1	complex	complex	ADJ
ejpam-6813	463	2	analysis	analysis	NOUN
ejpam-6813	463	3	and	and	CCONJ
ejpam-6813	463	4	operator	operator	NOUN
ejpam-6813	463	5	theory	theory	NOUN
ejpam-6813	463	6	,	,	PUNCT
ejpam-6813	463	7	19:221	19:221	NUM
ejpam-6813	463	8	,	,	PUNCT
ejpam-6813	463	9	2025	2025	NUM
ejpam-6813	463	10	.	.	PUNCT
ejpam-6813	464	1	r.	r.	PROPN
ejpam-6813	464	2	al	al	PROPN
ejpam-6813	464	3	-	-	PROPN
ejpam-6813	464	4	naimi	naimi	PROPN
ejpam-6813	464	5	et	et	PROPN
ejpam-6813	464	6	al	al	PROPN
ejpam-6813	464	7	.	.	PUNCT
ejpam-6813	464	8	/	/	SYM
ejpam-6813	464	9	eur	eur	PROPN
ejpam-6813	464	10	.	.	PUNCT
ejpam-6813	465	1	j.	j.	PROPN
ejpam-6813	465	2	pure	pure	PROPN
ejpam-6813	465	3	appl	appl	PROPN
ejpam-6813	465	4	.	.	PROPN
ejpam-6813	465	5	math	math	PROPN
ejpam-6813	465	6	,	,	PUNCT
ejpam-6813	465	7	18	18	NUM
ejpam-6813	465	8	(	(	PUNCT
ejpam-6813	465	9	4	4	NUM
ejpam-6813	465	10	)	)	PUNCT
ejpam-6813	465	11	(	(	PUNCT
ejpam-6813	465	12	2025	2025	NUM
ejpam-6813	465	13	)	)	PUNCT
ejpam-6813	465	14	,	,	PUNCT
ejpam-6813	465	15	6813	6813	NUM
ejpam-6813	465	16	15	15	NUM
ejpam-6813	465	17	of	of	ADP
ejpam-6813	465	18	15	15	NUM
ejpam-6813	466	1	[	[	SYM
ejpam-6813	466	2	15	15	NUM
ejpam-6813	466	3	]	]	X
ejpam-6813	466	4	r.	r.	PROPN
ejpam-6813	466	5	al	al	PROPN
ejpam-6813	466	6	-	-	PUNCT
ejpam-6813	466	7	naimi	naimi	PROPN
ejpam-6813	466	8	,	,	PUNCT
ejpam-6813	466	9	m.	m.	NOUN
ejpam-6813	466	10	al	al	PROPN
ejpam-6813	466	11	-	-	PUNCT
ejpam-6813	466	12	labadi	labadi	NOUN
ejpam-6813	466	13	,	,	PUNCT
ejpam-6813	466	14	and	and	CCONJ
ejpam-6813	466	15	w.	w.	PROPN
ejpam-6813	466	16	audeh	audeh	PROPN
ejpam-6813	466	17	.	.	PUNCT
ejpam-6813	467	1	generalizations	generalization	NOUN
ejpam-6813	467	2	of	of	ADP
ejpam-6813	467	3	p	p	PROPN
ejpam-6813	467	4	numerical	numerical	ADJ
ejpam-6813	467	5	radii	radius	NOUN
ejpam-6813	467	6	inequalities	inequality	NOUN
ejpam-6813	467	7	for	for	ADP
ejpam-6813	467	8	operators	operator	NOUN
ejpam-6813	467	9	.	.	PUNCT
ejpam-6813	468	1	jordan	jordan	PROPN
ejpam-6813	468	2	journal	journal	PROPN
ejpam-6813	468	3	of	of	ADP
ejpam-6813	468	4	mathematics	mathematics	PROPN
ejpam-6813	468	5	and	and	CCONJ
ejpam-6813	468	6	statistics	statistic	NOUN
ejpam-6813	468	7	,	,	PUNCT
ejpam-6813	468	8	18(3):421–431	18(3):421–431	PROPN
ejpam-6813	468	9	,	,	PUNCT
ejpam-6813	468	10	2025	2025	NUM
ejpam-6813	468	11	.	.	PUNCT