id	sid	tid	token	lemma	pos
ejpam-5009	1	1	european	european	PROPN
ejpam-5009	1	2	journal	journal	PROPN
ejpam-5009	1	3	of	of	ADP
ejpam-5009	1	4	pure	pure	ADJ
ejpam-5009	1	5	and	and	CCONJ
ejpam-5009	1	6	applied	apply	VERB
ejpam-5009	1	7	mathematics	mathematic	NOUN
ejpam-5009	1	8	vol	vol	NOUN
ejpam-5009	1	9	.	.	PROPN
ejpam-5009	2	1	17	17	NUM
ejpam-5009	2	2	,	,	PUNCT
ejpam-5009	2	3	no	no	INTJ
ejpam-5009	2	4	.	.	NOUN
ejpam-5009	2	5	1	1	NUM
ejpam-5009	2	6	,	,	PUNCT
ejpam-5009	2	7	2024	2024	NUM
ejpam-5009	2	8	,	,	PUNCT
ejpam-5009	2	9	243	243	NUM
ejpam-5009	2	10	-	-	SYM
ejpam-5009	2	11	247	247	NUM
ejpam-5009	2	12	issn	issn	PROPN
ejpam-5009	2	13	1307	1307	NUM
ejpam-5009	2	14	-	-	SYM
ejpam-5009	2	15	5543	5543	NUM
ejpam-5009	2	16	–	–	PUNCT
ejpam-5009	3	1	ejpam.com	ejpam.com	X
ejpam-5009	3	2	published	publish	VERB
ejpam-5009	3	3	by	by	ADP
ejpam-5009	3	4	new	new	PROPN
ejpam-5009	3	5	york	york	PROPN
ejpam-5009	3	6	business	business	PROPN
ejpam-5009	3	7	global	global	ADJ
ejpam-5009	3	8	matrix	matrix	NOUN
ejpam-5009	3	9	mixed	mixed	ADJ
ejpam-5009	3	10	inequalities	inequality	NOUN
ejpam-5009	3	11	feng	feng	PROPN
ejpam-5009	3	12	zhang1,∗	zhang1,∗	PROPN
ejpam-5009	3	13	,	,	PUNCT
ejpam-5009	3	14	chunwen	chunwen	NOUN
ejpam-5009	3	15	zhang2	zhang2	PROPN
ejpam-5009	3	16	1	1	NUM
ejpam-5009	3	17	department	department	NOUN
ejpam-5009	3	18	of	of	ADP
ejpam-5009	3	19	mathematics	mathematic	NOUN
ejpam-5009	3	20	,	,	PUNCT
ejpam-5009	3	21	southeast	southeast	ADJ
ejpam-5009	3	22	university	university	NOUN
ejpam-5009	3	23	chengxian	chengxian	PROPN
ejpam-5009	3	24	college	college	PROPN
ejpam-5009	3	25	,	,	PUNCT
ejpam-5009	3	26	nanjing	nanjing	PROPN
ejpam-5009	3	27	210000	210000	NUM
ejpam-5009	3	28	,	,	PUNCT
ejpam-5009	3	29	china	china	PROPN
ejpam-5009	3	30	2	2	NUM
ejpam-5009	3	31	department	department	NOUN
ejpam-5009	3	32	of	of	ADP
ejpam-5009	3	33	electrical	electrical	ADJ
ejpam-5009	3	34	and	and	CCONJ
ejpam-5009	3	35	computer	computer	NOUN
ejpam-5009	3	36	engineering	engineering	NOUN
ejpam-5009	3	37	,	,	PUNCT
ejpam-5009	3	38	southeast	southeast	PROPN
ejpam-5009	3	39	university	university	NOUN
ejpam-5009	3	40	chengxian	chengxian	PROPN
ejpam-5009	3	41	college	college	PROPN
ejpam-5009	3	42	,	,	PUNCT
ejpam-5009	3	43	nanjing	nanjing	PROPN
ejpam-5009	3	44	210000	210000	NUM
ejpam-5009	3	45	,	,	PUNCT
ejpam-5009	3	46	china	china	PROPN
ejpam-5009	3	47	abstract	abstract	NOUN
ejpam-5009	3	48	.	.	PUNCT
ejpam-5009	4	1	in	in	ADP
ejpam-5009	4	2	this	this	DET
ejpam-5009	4	3	paper	paper	NOUN
ejpam-5009	4	4	,	,	PUNCT
ejpam-5009	4	5	we	we	PRON
ejpam-5009	4	6	prove	prove	VERB
ejpam-5009	4	7	that	that	SCONJ
ejpam-5009	4	8	all	all	DET
ejpam-5009	4	9	the	the	DET
ejpam-5009	4	10	eigenvalues	eigenvalue	NOUN
ejpam-5009	4	11	of	of	ADP
ejpam-5009	4	12	arbitrarily	arbitrarily	ADV
ejpam-5009	4	13	complex	complex	ADJ
ejpam-5009	4	14	matrix	matrix	NOUN
ejpam-5009	4	15	are	be	AUX
ejpam-5009	4	16	located	locate	VERB
ejpam-5009	4	17	in	in	ADP
ejpam-5009	4	18	one	one	NUM
ejpam-5009	4	19	closed	closed	ADJ
ejpam-5009	4	20	disk	disk	NOUN
ejpam-5009	4	21	,	,	PUNCT
ejpam-5009	4	22	which	which	PRON
ejpam-5009	4	23	is	be	AUX
ejpam-5009	4	24	a	a	DET
ejpam-5009	4	25	refinement	refinement	NOUN
ejpam-5009	4	26	of	of	ADP
ejpam-5009	4	27	some	some	DET
ejpam-5009	4	28	existing	exist	VERB
ejpam-5009	4	29	inequalities	inequality	NOUN
ejpam-5009	4	30	.	.	PUNCT
ejpam-5009	5	1	2020	2020	NUM
ejpam-5009	5	2	mathematics	mathematic	NOUN
ejpam-5009	5	3	subject	subject	NOUN
ejpam-5009	5	4	classifications	classification	NOUN
ejpam-5009	5	5	:	:	PUNCT
ejpam-5009	5	6	47a30	47a30	NUM
ejpam-5009	5	7	,	,	PUNCT
ejpam-5009	5	8	15a42	15a42	NUM
ejpam-5009	5	9	,	,	PUNCT
ejpam-5009	5	10	15a18	15a18	NUM
ejpam-5009	5	11	key	key	ADJ
ejpam-5009	5	12	words	word	NOUN
ejpam-5009	5	13	and	and	CCONJ
ejpam-5009	5	14	phrases	phrase	NOUN
ejpam-5009	5	15	:	:	PUNCT
ejpam-5009	5	16	disk	disk	NOUN
ejpam-5009	5	17	,	,	PUNCT
ejpam-5009	5	18	majorization	majorization	NOUN
ejpam-5009	5	19	,	,	PUNCT
ejpam-5009	5	20	unitarily	unitarily	ADV
ejpam-5009	5	21	invariant	invariant	ADJ
ejpam-5009	5	22	norms	norm	NOUN
ejpam-5009	5	23	,	,	PUNCT
ejpam-5009	5	24	eigenvalue	eigenvalue	NOUN
ejpam-5009	5	25	;	;	PUNCT
ejpam-5009	5	26	location	location	NOUN
ejpam-5009	5	27	1	1	NUM
ejpam-5009	5	28	.	.	PUNCT
ejpam-5009	6	1	introduction	introduction	NOUN
ejpam-5009	6	2	we	we	PRON
ejpam-5009	6	3	denote	denote	VERB
ejpam-5009	6	4	by	by	ADP
ejpam-5009	6	5	mn	mn	PROPN
ejpam-5009	6	6	the	the	DET
ejpam-5009	6	7	vector	vector	NOUN
ejpam-5009	6	8	space	space	NOUN
ejpam-5009	6	9	of	of	ADP
ejpam-5009	6	10	all	all	DET
ejpam-5009	6	11	complex	complex	ADJ
ejpam-5009	6	12	n×	n×	NOUN
ejpam-5009	6	13	n	n	NOUN
ejpam-5009	6	14	matrices	matrix	NOUN
ejpam-5009	6	15	.	.	PUNCT
ejpam-5009	7	1	the	the	DET
ejpam-5009	7	2	notation	notation	NOUN
ejpam-5009	7	3	a	a	DET
ejpam-5009	7	4	≥	≥	NOUN
ejpam-5009	7	5	0	0	NUM
ejpam-5009	7	6	is	be	AUX
ejpam-5009	7	7	used	use	VERB
ejpam-5009	7	8	to	to	PART
ejpam-5009	7	9	mean	mean	VERB
ejpam-5009	7	10	that	that	SCONJ
ejpam-5009	7	11	a	a	PRON
ejpam-5009	7	12	is	be	AUX
ejpam-5009	7	13	positive	positive	ADJ
ejpam-5009	7	14	semidefinite	semidefinite	NOUN
ejpam-5009	7	15	.	.	PUNCT
ejpam-5009	8	1	for	for	ADP
ejpam-5009	8	2	a	a	DET
ejpam-5009	8	3	∈	∈	PROPN
ejpam-5009	8	4	mn	mn	PROPN
ejpam-5009	8	5	,	,	PUNCT
ejpam-5009	8	6	the	the	DET
ejpam-5009	8	7	conjugate	conjugate	ADJ
ejpam-5009	8	8	transpose	transpose	NOUN
ejpam-5009	8	9	of	of	ADP
ejpam-5009	8	10	a	a	PRON
ejpam-5009	8	11	is	be	AUX
ejpam-5009	8	12	denoted	denote	VERB
ejpam-5009	8	13	by	by	ADP
ejpam-5009	8	14	a∗.	a∗.	NOUN
ejpam-5009	8	15	denote	denote	VERB
ejpam-5009	8	16	by	by	ADP
ejpam-5009	8	17	λj	λj	PROPN
ejpam-5009	8	18	(	(	PUNCT
ejpam-5009	8	19	a	a	NOUN
ejpam-5009	8	20	)	)	PUNCT
ejpam-5009	8	21	(	(	PUNCT
ejpam-5009	8	22	1	1	NUM
ejpam-5009	8	23	≤	≤	NUM
ejpam-5009	8	24	j	j	PROPN
ejpam-5009	8	25	≤	≤	NUM
ejpam-5009	8	26	n	n	CCONJ
ejpam-5009	8	27	)	)	PUNCT
ejpam-5009	8	28	the	the	DET
ejpam-5009	8	29	class	class	NOUN
ejpam-5009	8	30	of	of	ADP
ejpam-5009	8	31	all	all	DET
ejpam-5009	8	32	eigenvalues	eigenvalue	NOUN
ejpam-5009	8	33	of	of	ADP
ejpam-5009	8	34	a	a	DET
ejpam-5009	8	35	∈	∈	PROPN
ejpam-5009	8	36	mn	mn	PROPN
ejpam-5009	8	37	and	and	CCONJ
ejpam-5009	8	38	∥a∥f	∥a∥f	NOUN
ejpam-5009	8	39	=	=	SYM
ejpam-5009	8	40	√	√	NOUN
ejpam-5009	8	41	tr	tr	PRON
ejpam-5009	8	42	(	(	PUNCT
ejpam-5009	8	43	aa∗	aa∗	NOUN
ejpam-5009	8	44	)	)	PUNCT
ejpam-5009	8	45	,	,	PUNCT
ejpam-5009	9	1	[	[	X
ejpam-5009	9	2	a	a	X
ejpam-5009	9	3	,	,	PUNCT
ejpam-5009	9	4	b	b	NOUN
ejpam-5009	9	5	]	]	X
ejpam-5009	9	6	=	=	SYM
ejpam-5009	9	7	ab	ab	PROPN
ejpam-5009	9	8	−	−	PROPN
ejpam-5009	9	9	ba	ba	PROPN
ejpam-5009	9	10	.	.	PUNCT
ejpam-5009	10	1	the	the	DET
ejpam-5009	10	2	singular	singular	ADJ
ejpam-5009	10	3	values	value	NOUN
ejpam-5009	10	4	of	of	ADP
ejpam-5009	10	5	a	a	PRON
ejpam-5009	10	6	are	be	AUX
ejpam-5009	10	7	enumerated	enumerate	VERB
ejpam-5009	10	8	as	as	ADP
ejpam-5009	10	9	s1	s1	NOUN
ejpam-5009	10	10	(	(	PUNCT
ejpam-5009	10	11	a	a	PRON
ejpam-5009	10	12	)	)	PUNCT
ejpam-5009	10	13	≥	≥	NOUN
ejpam-5009	10	14	s2	s2	NOUN
ejpam-5009	10	15	(	(	PUNCT
ejpam-5009	10	16	a	a	NOUN
ejpam-5009	10	17	)	)	PUNCT
ejpam-5009	10	18	≥	≥	NOUN
ejpam-5009	10	19	·	·	PUNCT
ejpam-5009	10	20	·	·	PUNCT
ejpam-5009	10	21	·	·	PUNCT
ejpam-5009	10	22	≥	≥	X
ejpam-5009	11	1	sn	sn	INTJ
ejpam-5009	11	2	(	(	PUNCT
ejpam-5009	11	3	a	a	NOUN
ejpam-5009	11	4	)	)	PUNCT
ejpam-5009	11	5	.	.	PUNCT
ejpam-5009	12	1	these	these	PRON
ejpam-5009	12	2	are	be	AUX
ejpam-5009	12	3	the	the	DET
ejpam-5009	12	4	eigenvalues	eigenvalue	NOUN
ejpam-5009	12	5	of	of	ADP
ejpam-5009	12	6	the	the	DET
ejpam-5009	12	7	positive	positive	ADJ
ejpam-5009	12	8	semidefinite	semidefinite	NOUN
ejpam-5009	12	9	matrix	matrix	NOUN
ejpam-5009	12	10	|a|	|a|	PROPN
ejpam-5009	12	11	=	=	SYM
ejpam-5009	12	12	(	(	PUNCT
ejpam-5009	12	13	a∗a	a∗a	X
ejpam-5009	12	14	)	)	PUNCT
ejpam-5009	12	15	1	1	NUM
ejpam-5009	12	16	2	2	NUM
ejpam-5009	12	17	.	.	PUNCT
ejpam-5009	13	1	the	the	DET
ejpam-5009	13	2	estimation	estimation	NOUN
ejpam-5009	13	3	and	and	CCONJ
ejpam-5009	13	4	location	location	NOUN
ejpam-5009	13	5	of	of	ADP
ejpam-5009	13	6	eigenvalues	eigenvalue	NOUN
ejpam-5009	13	7	are	be	AUX
ejpam-5009	13	8	always	always	ADV
ejpam-5009	13	9	hot	hot	ADJ
ejpam-5009	13	10	topics	topic	NOUN
ejpam-5009	13	11	of	of	ADP
ejpam-5009	13	12	matrix	matrix	NOUN
ejpam-5009	13	13	analysis	analysis	NOUN
ejpam-5009	13	14	[	[	X
ejpam-5009	13	15	1	1	NUM
ejpam-5009	13	16	]	]	PUNCT
ejpam-5009	13	17	,	,	PUNCT
ejpam-5009	13	18	[	[	X
ejpam-5009	13	19	2	2	NUM
ejpam-5009	13	20	]	]	PUNCT
ejpam-5009	13	21	.	.	PUNCT
ejpam-5009	14	1	it	it	PRON
ejpam-5009	14	2	plays	play	VERB
ejpam-5009	14	3	an	an	DET
ejpam-5009	14	4	important	important	ADJ
ejpam-5009	14	5	role	role	NOUN
ejpam-5009	14	6	in	in	ADP
ejpam-5009	14	7	many	many	ADJ
ejpam-5009	14	8	fields	field	NOUN
ejpam-5009	14	9	of	of	ADP
ejpam-5009	14	10	applied	apply	VERB
ejpam-5009	14	11	science	science	NOUN
ejpam-5009	14	12	.	.	PUNCT
ejpam-5009	15	1	let	let	VERB
ejpam-5009	15	2	m	m	PRON
ejpam-5009	15	3	∈	∈	PROPN
ejpam-5009	15	4	mn	mn	PROPN
ejpam-5009	15	5	be	be	AUX
ejpam-5009	15	6	an	an	DET
ejpam-5009	15	7	n×n	n×n	PROPN
ejpam-5009	15	8	complex	complex	ADJ
ejpam-5009	15	9	matrix	matrix	NOUN
ejpam-5009	15	10	partitioned	partition	VERB
ejpam-5009	15	11	as	as	ADP
ejpam-5009	15	12	m	m	NOUN
ejpam-5009	15	13	=	=	PUNCT
ejpam-5009	15	14	[	[	PUNCT
ejpam-5009	15	15	ak	ak	PROPN
ejpam-5009	15	16	bk	bk	PROPN
ejpam-5009	15	17	,	,	PUNCT
ejpam-5009	15	18	n−k	n−k	NOUN
ejpam-5009	15	19	cn−k	cn−k	NOUN
ejpam-5009	15	20	,	,	PUNCT
ejpam-5009	15	21	k	k	PROPN
ejpam-5009	15	22	dn−k	dn−k	PROPN
ejpam-5009	15	23	]	]	PUNCT
ejpam-5009	15	24	,	,	PUNCT
ejpam-5009	15	25	where	where	SCONJ
ejpam-5009	15	26	1	1	NUM
ejpam-5009	15	27	≤	≤	NUM
ejpam-5009	15	28	k	k	X
ejpam-5009	15	29	≤	≤	PROPN
ejpam-5009	15	30	n−	n−	PROPN
ejpam-5009	15	31	1	1	NUM
ejpam-5009	15	32	.	.	PUNCT
ejpam-5009	16	1	the	the	DET
ejpam-5009	16	2	following	follow	VERB
ejpam-5009	16	3	estimation	estimation	NOUN
ejpam-5009	16	4	n∑	n∑	PROPN
ejpam-5009	16	5	i=1	i=1	PROPN
ejpam-5009	16	6	|λi|2	|λi|2	PROPN
ejpam-5009	16	7	≤	≤	PROPN
ejpam-5009	16	8	∥m∥2f	∥m∥2f	PUNCT
ejpam-5009	16	9	−	−	PROPN
ejpam-5009	16	10	max	max	PROPN
ejpam-5009	16	11	1≤k≤n−1	1≤k≤n−1	NUM
ejpam-5009	16	12	(	(	PUNCT
ejpam-5009	16	13	∥bk	∥bk	PROPN
ejpam-5009	16	14	,	,	PUNCT
ejpam-5009	16	15	n−k∥f	n−k∥f	NOUN
ejpam-5009	16	16	−	−	PROPN
ejpam-5009	16	17	∥cn−k	∥cn−k	PROPN
ejpam-5009	16	18	,	,	PUNCT
ejpam-5009	16	19	k∥f	k∥f	NOUN
ejpam-5009	16	20	)	)	PUNCT
ejpam-5009	16	21	2	2	NUM
ejpam-5009	16	22	is	be	AUX
ejpam-5009	16	23	an	an	DET
ejpam-5009	16	24	elegant	elegant	ADJ
ejpam-5009	16	25	result	result	NOUN
ejpam-5009	16	26	on	on	ADP
ejpam-5009	16	27	eigenvalues	eigenvalue	NOUN
ejpam-5009	16	28	due	due	ADP
ejpam-5009	16	29	to	to	ADP
ejpam-5009	16	30	tu	tu	PROPN
ejpam-5009	16	31	[	[	X
ejpam-5009	16	32	3	3	NUM
ejpam-5009	16	33	]	]	PUNCT
ejpam-5009	16	34	.	.	PUNCT
ejpam-5009	17	1	∗corresponding	∗corresponde	VERB
ejpam-5009	17	2	author	author	NOUN
ejpam-5009	17	3	.	.	PUNCT
ejpam-5009	18	1	doi	doi	NOUN
ejpam-5009	18	2	:	:	PUNCT
ejpam-5009	18	3	https://doi.org/10.29020/nybg.ejpam.v17i1.5009	https://doi.org/10.29020/nybg.ejpam.v17i1.5009	PROPN
ejpam-5009	18	4	email	email	NOUN
ejpam-5009	18	5	addresses	address	NOUN
ejpam-5009	18	6	:	:	PUNCT
ejpam-5009	18	7	fzhang1024@163.com	fzhang1024@163.com	X
ejpam-5009	18	8	(	(	PUNCT
ejpam-5009	18	9	f.	f.	PROPN
ejpam-5009	18	10	zhang	zhang	PROPN
ejpam-5009	18	11	)	)	PUNCT
ejpam-5009	18	12	,	,	PUNCT
ejpam-5009	18	13	2153228327@qq.com	2153228327@qq.com	NUM
ejpam-5009	18	14	(	(	PUNCT
ejpam-5009	18	15	c.	c.	PROPN
ejpam-5009	18	16	zhang	zhang	PROPN
ejpam-5009	18	17	)	)	PUNCT
ejpam-5009	18	18	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5009	19	1	243	243	NUM
ejpam-5009	19	2	©	©	ADP
ejpam-5009	19	3	2024	2024	NUM
ejpam-5009	19	4	ejpam	ejpam	NOUN
ejpam-5009	19	5	all	all	DET
ejpam-5009	19	6	rights	right	NOUN
ejpam-5009	19	7	reserved	reserve	VERB
ejpam-5009	19	8	.	.	PUNCT
ejpam-5009	20	1	f.	f.	PROPN
ejpam-5009	20	2	zhang	zhang	PROPN
ejpam-5009	20	3	,	,	PUNCT
ejpam-5009	20	4	c.	c.	PROPN
ejpam-5009	20	5	zhang	zhang	PROPN
ejpam-5009	20	6	/	/	SYM
ejpam-5009	20	7	eur	eur	PROPN
ejpam-5009	20	8	.	.	PUNCT
ejpam-5009	21	1	j.	j.	PROPN
ejpam-5009	21	2	pure	pure	PROPN
ejpam-5009	21	3	appl	appl	PROPN
ejpam-5009	21	4	.	.	PROPN
ejpam-5009	21	5	math	math	PROPN
ejpam-5009	21	6	,	,	PUNCT
ejpam-5009	21	7	17	17	NUM
ejpam-5009	21	8	(	(	PUNCT
ejpam-5009	21	9	1	1	NUM
ejpam-5009	21	10	)	)	PUNCT
ejpam-5009	21	11	(	(	PUNCT
ejpam-5009	21	12	2024	2024	NUM
ejpam-5009	21	13	)	)	PUNCT
ejpam-5009	21	14	,	,	PUNCT
ejpam-5009	21	15	243	243	NUM
ejpam-5009	21	16	-	-	SYM
ejpam-5009	21	17	247	247	NUM
ejpam-5009	21	18	244	244	NUM
ejpam-5009	21	19	in	in	ADP
ejpam-5009	21	20	[	[	X
ejpam-5009	21	21	4	4	NUM
ejpam-5009	21	22	]	]	PUNCT
ejpam-5009	21	23	,	,	PUNCT
ejpam-5009	21	24	gu	gu	PROPN
ejpam-5009	21	25	proposed	propose	VERB
ejpam-5009	21	26	a	a	DET
ejpam-5009	21	27	new	new	ADJ
ejpam-5009	21	28	idea	idea	NOUN
ejpam-5009	21	29	which	which	PRON
ejpam-5009	21	30	uses	use	VERB
ejpam-5009	21	31	only	only	ADV
ejpam-5009	21	32	one	one	NUM
ejpam-5009	21	33	single	single	ADJ
ejpam-5009	21	34	closed	closed	ADJ
ejpam-5009	21	35	disk	disk	NOUN
ejpam-5009	21	36	to	to	PART
ejpam-5009	21	37	locate	locate	VERB
ejpam-5009	21	38	eigenvalues	eigenvalue	NOUN
ejpam-5009	21	39	of	of	ADP
ejpam-5009	21	40	a	a	DET
ejpam-5009	21	41	given	give	VERB
ejpam-5009	21	42	n×n	n×n	PROPN
ejpam-5009	21	43	complex	complex	ADJ
ejpam-5009	21	44	matrix	matrix	NOUN
ejpam-5009	21	45	.	.	PUNCT
ejpam-5009	22	1	he	he	PRON
ejpam-5009	22	2	proved	prove	VERB
ejpam-5009	22	3	that	that	SCONJ
ejpam-5009	22	4	all	all	DET
ejpam-5009	22	5	the	the	DET
ejpam-5009	22	6	eigenvalues	eigenvalue	NOUN
ejpam-5009	22	7	of	of	ADP
ejpam-5009	22	8	any	any	DET
ejpam-5009	22	9	complex	complex	ADJ
ejpam-5009	22	10	matrix	matrix	NOUN
ejpam-5009	22	11	a	a	PRON
ejpam-5009	22	12	are	be	AUX
ejpam-5009	22	13	located	locate	VERB
ejpam-5009	22	14	in	in	ADP
ejpam-5009	22	15	the	the	DET
ejpam-5009	22	16	following	follow	VERB
ejpam-5009	22	17	disk:∣∣∣∣λj	disk:∣∣∣∣λj	NOUN
ejpam-5009	22	18	−	−	PROPN
ejpam-5009	22	19	tra	tra	NOUN
ejpam-5009	22	20	n	n	PROPN
ejpam-5009	22	21	∣∣∣∣	∣∣∣∣	NOUN
ejpam-5009	22	22	≤	≤	NOUN
ejpam-5009	22	23	(	(	PUNCT
ejpam-5009	22	24	n−	n−	NOUN
ejpam-5009	22	25	1	1	NUM
ejpam-5009	22	26	n	n	NOUN
ejpam-5009	22	27	(	(	PUNCT
ejpam-5009	22	28	∥a∥2f	∥a∥2f	PROPN
ejpam-5009	22	29	−	−	NOUN
ejpam-5009	22	30	|tra|2	|tra|2	PROPN
ejpam-5009	22	31	n	n	NOUN
ejpam-5009	22	32	)	)	PUNCT
ejpam-5009	22	33	)	)	PUNCT
ejpam-5009	22	34	1	1	NUM
ejpam-5009	22	35	2	2	NUM
ejpam-5009	22	36	(	(	PUNCT
ejpam-5009	22	37	1	1	NUM
ejpam-5009	22	38	)	)	PUNCT
ejpam-5009	22	39	for	for	ADP
ejpam-5009	22	40	j	j	PROPN
ejpam-5009	22	41	=	=	SYM
ejpam-5009	22	42	1	1	NUM
ejpam-5009	22	43	,	,	PUNCT
ejpam-5009	22	44	2	2	NUM
ejpam-5009	22	45	,	,	PUNCT
ejpam-5009	22	46	·	·	PUNCT
ejpam-5009	22	47	·	·	PUNCT
ejpam-5009	22	48	·	·	PUNCT
ejpam-5009	22	49	,	,	PUNCT
ejpam-5009	22	50	n.	n.	NOUN
ejpam-5009	22	51	zou	zou	PROPN
ejpam-5009	22	52	et	et	PROPN
ejpam-5009	22	53	al	al	PROPN
ejpam-5009	22	54	.	.	PUNCT
ejpam-5009	23	1	[	[	X
ejpam-5009	23	2	5	5	NUM
ejpam-5009	23	3	]	]	PUNCT
ejpam-5009	23	4	showed	show	VERB
ejpam-5009	23	5	that	that	SCONJ
ejpam-5009	23	6	all	all	DET
ejpam-5009	23	7	eigenvalues	eigenvalue	NOUN
ejpam-5009	23	8	of	of	ADP
ejpam-5009	23	9	m	m	NOUN
ejpam-5009	23	10	are	be	AUX
ejpam-5009	23	11	located	locate	VERB
ejpam-5009	23	12	in	in	ADP
ejpam-5009	23	13	the	the	DET
ejpam-5009	23	14	following	follow	VERB
ejpam-5009	23	15	disk	disk	NOUN
ejpam-5009	23	16	:	:	PUNCT
ejpam-5009	23	17	{	{	PUNCT
ejpam-5009	23	18	z	z	NOUN
ejpam-5009	23	19	∈	∈	PROPN
ejpam-5009	23	20	c	c	NOUN
ejpam-5009	23	21	:	:	PUNCT
ejpam-5009	23	22	∣∣∣∣z	∣∣∣∣z	NOUN
ejpam-5009	23	23	−	−	PROPN
ejpam-5009	23	24	trm	trm	PROPN
ejpam-5009	23	25	n	n	PRON
ejpam-5009	23	26	∣∣∣∣	∣∣∣∣	NOUN
ejpam-5009	23	27	≤	≤	NOUN
ejpam-5009	23	28	√	√	ADP
ejpam-5009	23	29	∥m∥2f	∥m∥2f	PROPN
ejpam-5009	23	30	−	−	PROPN
ejpam-5009	23	31	|trm	|trm	ADJ
ejpam-5009	23	32	|2	|2	NUM
ejpam-5009	23	33	n	n	CCONJ
ejpam-5009	23	34	−	−	PROPN
ejpam-5009	23	35	max	max	PROPN
ejpam-5009	23	36	1≤k≤n−1	1≤k≤n−1	NUM
ejpam-5009	23	37	(	(	PUNCT
ejpam-5009	23	38	∥bk	∥bk	PROPN
ejpam-5009	23	39	,	,	PUNCT
ejpam-5009	23	40	n−k∥f	n−k∥f	NOUN
ejpam-5009	24	1	−	−	PROPN
ejpam-5009	24	2	∥cn−k	∥cn−k	PROPN
ejpam-5009	24	3	,	,	PUNCT
ejpam-5009	24	4	k∥f	k∥f	NOUN
ejpam-5009	24	5	)	)	PUNCT
ejpam-5009	24	6	2	2	NUM
ejpam-5009	24	7	}	}	PUNCT
ejpam-5009	24	8	.	.	PUNCT
ejpam-5009	25	1	(	(	PUNCT
ejpam-5009	25	2	2	2	X
ejpam-5009	25	3	)	)	PUNCT
ejpam-5009	25	4	let	let	VERB
ejpam-5009	25	5	m	m	PRON
ejpam-5009	25	6	(	(	PUNCT
ejpam-5009	25	7	x	x	NOUN
ejpam-5009	25	8	)	)	PUNCT
ejpam-5009	25	9	=	=	SYM
ejpam-5009	25	10	[	[	PUNCT
ejpam-5009	25	11	ak	ak	PROPN
ejpam-5009	25	12	xbk	xbk	PROPN
ejpam-5009	25	13	,	,	PUNCT
ejpam-5009	25	14	n−k	n−k	PROPN
ejpam-5009	25	15	x−1cn−k	x−1cn−k	PROPN
ejpam-5009	25	16	,	,	PUNCT
ejpam-5009	25	17	k	k	PROPN
ejpam-5009	25	18	dn−k	dn−k	PROPN
ejpam-5009	25	19	]	]	PUNCT
ejpam-5009	25	20	,	,	PUNCT
ejpam-5009	25	21	where	where	SCONJ
ejpam-5009	25	22	ak	ak	PROPN
ejpam-5009	25	23	is	be	AUX
ejpam-5009	25	24	a	a	DET
ejpam-5009	25	25	k	k	PROPN
ejpam-5009	25	26	×	×	PROPN
ejpam-5009	25	27	k	k	PROPN
ejpam-5009	25	28	principal	principal	ADJ
ejpam-5009	25	29	submatrix	submatrix	NOUN
ejpam-5009	25	30	of	of	ADP
ejpam-5009	25	31	m	m	PROPN
ejpam-5009	25	32	(	(	PUNCT
ejpam-5009	25	33	1	1	NUM
ejpam-5009	25	34	≤	≤	NUM
ejpam-5009	25	35	k	k	X
ejpam-5009	25	36	≤	≤	PROPN
ejpam-5009	25	37	n−	n−	PROPN
ejpam-5009	25	38	1	1	NUM
ejpam-5009	25	39	)	)	PUNCT
ejpam-5009	25	40	and	and	CCONJ
ejpam-5009	25	41	x	x	X
ejpam-5009	25	42	is	be	AUX
ejpam-5009	25	43	any	any	DET
ejpam-5009	25	44	non	non	ADJ
ejpam-5009	25	45	-	-	ADJ
ejpam-5009	25	46	zero	zero	ADJ
ejpam-5009	25	47	real	real	ADJ
ejpam-5009	25	48	number	number	NOUN
ejpam-5009	25	49	.	.	PUNCT
ejpam-5009	26	1	for	for	ADP
ejpam-5009	26	2	convenience	convenience	NOUN
ejpam-5009	26	3	,	,	PUNCT
ejpam-5009	26	4	we	we	PRON
ejpam-5009	26	5	write	write	VERB
ejpam-5009	26	6	,	,	PUNCT
ejpam-5009	26	7	respectively	respectively	ADV
ejpam-5009	26	8	.	.	PUNCT
ejpam-5009	27	1	△	△	PROPN
ejpam-5009	27	2	m	m	PROPN
ejpam-5009	27	3	(	(	PUNCT
ejpam-5009	27	4	k	k	X
ejpam-5009	27	5	,	,	PUNCT
ejpam-5009	27	6	x	x	NOUN
ejpam-5009	27	7	)	)	PUNCT
ejpam-5009	27	8	=	=	SYM
ejpam-5009	27	9	∥m∥2f	∥m∥2f	NUM
ejpam-5009	28	1	−	−	PUNCT
ejpam-5009	29	1	[	[	X
ejpam-5009	29	2	(	(	PUNCT
ejpam-5009	29	3	1−	1−	NUM
ejpam-5009	29	4	x2	x2	PROPN
ejpam-5009	29	5	)	)	PUNCT
ejpam-5009	29	6	∥bk	∥bk	ADP
ejpam-5009	29	7	,	,	PUNCT
ejpam-5009	29	8	n−k∥2f	n−k∥2f	NOUN
ejpam-5009	29	9	+	+	CCONJ
ejpam-5009	29	10	(	(	PUNCT
ejpam-5009	29	11	1−	1−	NUM
ejpam-5009	29	12	x−2	x−2	PROPN
ejpam-5009	29	13	)	)	PUNCT
ejpam-5009	29	14	∥cn−k	∥cn−k	PROPN
ejpam-5009	29	15	,	,	PUNCT
ejpam-5009	29	16	k∥2f	k∥2f	NOUN
ejpam-5009	29	17	]	]	PUNCT
ejpam-5009	29	18	−	−	PROPN
ejpam-5009	29	19	|trm	|trm	ADJ
ejpam-5009	29	20	|2	|2	NUM
ejpam-5009	29	21	n	n	NOUN
ejpam-5009	29	22	and	and	CCONJ
ejpam-5009	29	23	fm	fm	PROPN
ejpam-5009	29	24	(	(	PUNCT
ejpam-5009	29	25	k	k	X
ejpam-5009	29	26	,	,	PUNCT
ejpam-5009	29	27	x	x	NOUN
ejpam-5009	29	28	)	)	PUNCT
ejpam-5009	29	29	=	=	SYM
ejpam-5009	29	30	(	(	PUNCT
ejpam-5009	29	31	(	(	PUNCT
ejpam-5009	29	32	△	△	X
ejpam-5009	29	33	m	m	PROPN
ejpam-5009	29	34	(	(	PUNCT
ejpam-5009	29	35	k	k	X
ejpam-5009	29	36	,	,	PUNCT
ejpam-5009	29	37	x))2	x))2	PROPN
ejpam-5009	30	1	−	−	NOUN
ejpam-5009	30	2	1	1	NUM
ejpam-5009	30	3	2	2	NUM
ejpam-5009	30	4	∥[m	∥[m	NOUN
ejpam-5009	30	5	(	(	PUNCT
ejpam-5009	30	6	x	x	NOUN
ejpam-5009	30	7	)	)	PUNCT
ejpam-5009	30	8	,	,	PUNCT
ejpam-5009	30	9	m	m	PROPN
ejpam-5009	30	10	(	(	PUNCT
ejpam-5009	30	11	x)∗]∥2f	x)∗]∥2f	PROPN
ejpam-5009	30	12	)	)	PUNCT
ejpam-5009	31	1	1	1	NUM
ejpam-5009	31	2	2	2	NUM
ejpam-5009	31	3	+	+	NUM
ejpam-5009	31	4	|trm	|trm	ADJ
ejpam-5009	31	5	|2	|2	NUM
ejpam-5009	31	6	n	n	NOUN
ejpam-5009	31	7	.	.	PUNCT
ejpam-5009	32	1	in	in	ADP
ejpam-5009	32	2	[	[	X
ejpam-5009	32	3	6	6	NUM
ejpam-5009	32	4	]	]	PUNCT
ejpam-5009	32	5	,	,	PUNCT
ejpam-5009	32	6	wu	wu	PROPN
ejpam-5009	32	7	et	et	PROPN
ejpam-5009	32	8	al	al	PROPN
ejpam-5009	32	9	.	.	PROPN
ejpam-5009	32	10	proved	prove	VERB
ejpam-5009	32	11	that∣∣∣∣λj(m)−	that∣∣∣∣λj(m)−	PROPN
ejpam-5009	32	12	trm	trm	PROPN
ejpam-5009	32	13	n	n	CCONJ
ejpam-5009	32	14	∣∣∣∣	∣∣∣∣	NOUN
ejpam-5009	32	15	≤	≤	NUM
ejpam-5009	32	16	min	min	NOUN
ejpam-5009	33	1	x̸=0	x̸=0	PROPN
ejpam-5009	33	2	min	min	PROPN
ejpam-5009	33	3	1≤k≤n−1	1≤k≤n−1	NUM
ejpam-5009	33	4	√	√	NUM
ejpam-5009	33	5	n−	n−	NOUN
ejpam-5009	33	6	1	1	NUM
ejpam-5009	33	7	n	n	PROPN
ejpam-5009	33	8	(	(	PUNCT
ejpam-5009	33	9	fm	fm	PROPN
ejpam-5009	33	10	(	(	PUNCT
ejpam-5009	33	11	k	k	X
ejpam-5009	33	12	,	,	PUNCT
ejpam-5009	33	13	x)−	x)−	PROPN
ejpam-5009	33	14	|trm	|trm	ADJ
ejpam-5009	33	15	|2	|2	NUM
ejpam-5009	33	16	n	n	CCONJ
ejpam-5009	33	17	)	)	PUNCT
ejpam-5009	33	18	1	1	NUM
ejpam-5009	33	19	2	2	NUM
ejpam-5009	33	20	,	,	PUNCT
ejpam-5009	33	21	(	(	PUNCT
ejpam-5009	33	22	3	3	X
ejpam-5009	33	23	)	)	PUNCT
ejpam-5009	33	24	which	which	PRON
ejpam-5009	33	25	is	be	AUX
ejpam-5009	33	26	a	a	DET
ejpam-5009	33	27	refinement	refinement	NOUN
ejpam-5009	33	28	of	of	ADP
ejpam-5009	33	29	inequality	inequality	NOUN
ejpam-5009	33	30	(	(	PUNCT
ejpam-5009	33	31	2	2	NUM
ejpam-5009	33	32	)	)	PUNCT
ejpam-5009	33	33	.	.	PUNCT
ejpam-5009	34	1	it	it	PRON
ejpam-5009	34	2	is	be	AUX
ejpam-5009	34	3	natural	natural	ADJ
ejpam-5009	34	4	to	to	PART
ejpam-5009	34	5	ask	ask	VERB
ejpam-5009	34	6	whether	whether	SCONJ
ejpam-5009	34	7	stronger	strong	ADJ
ejpam-5009	34	8	inequality	inequality	NOUN
ejpam-5009	34	9	of	of	ADP
ejpam-5009	34	10	(	(	PUNCT
ejpam-5009	34	11	2	2	X
ejpam-5009	34	12	)	)	PUNCT
ejpam-5009	34	13	might	might	AUX
ejpam-5009	34	14	be	be	AUX
ejpam-5009	34	15	proved	prove	VERB
ejpam-5009	34	16	.	.	PUNCT
ejpam-5009	35	1	this	this	PRON
ejpam-5009	35	2	is	be	AUX
ejpam-5009	35	3	a	a	DET
ejpam-5009	35	4	part	part	NOUN
ejpam-5009	35	5	of	of	ADP
ejpam-5009	35	6	the	the	DET
ejpam-5009	35	7	motivation	motivation	NOUN
ejpam-5009	35	8	for	for	ADP
ejpam-5009	35	9	our	our	PRON
ejpam-5009	35	10	study	study	NOUN
ejpam-5009	35	11	.	.	PUNCT
ejpam-5009	36	1	2	2	X
ejpam-5009	36	2	.	.	X
ejpam-5009	36	3	main	main	ADJ
ejpam-5009	36	4	result	result	NOUN
ejpam-5009	36	5	we	we	PRON
ejpam-5009	36	6	let	let	VERB
ejpam-5009	36	7	the	the	DET
ejpam-5009	36	8	symbol	symbol	NOUN
ejpam-5009	37	1	sl	sl	NOUN
ejpam-5009	37	2	denote	denote	VERB
ejpam-5009	37	3	the	the	DET
ejpam-5009	37	4	set	set	NOUN
ejpam-5009	37	5	{	{	PUNCT
ejpam-5009	37	6	1	1	NUM
ejpam-5009	37	7	,	,	PUNCT
ejpam-5009	37	8	·	·	PUNCT
ejpam-5009	37	9	·	·	PUNCT
ejpam-5009	37	10	·	·	PUNCT
ejpam-5009	37	11	n}\{l	n}\{l	NOUN
ejpam-5009	37	12	}	}	PUNCT
ejpam-5009	37	13	for	for	ADP
ejpam-5009	37	14	l	l	NOUN
ejpam-5009	37	15	=	=	SYM
ejpam-5009	37	16	1	1	NUM
ejpam-5009	37	17	,	,	PUNCT
ejpam-5009	37	18	2	2	NUM
ejpam-5009	37	19	,	,	PUNCT
ejpam-5009	37	20	·	·	PUNCT
ejpam-5009	37	21	·	·	PUNCT
ejpam-5009	37	22	·	·	PUNCT
ejpam-5009	37	23	,	,	PUNCT
ejpam-5009	37	24	n.	n.	NOUN
ejpam-5009	37	25	in	in	ADP
ejpam-5009	37	26	this	this	DET
ejpam-5009	37	27	section	section	NOUN
ejpam-5009	37	28	,	,	PUNCT
ejpam-5009	37	29	a	a	DET
ejpam-5009	37	30	sharper	sharp	ADJ
ejpam-5009	37	31	estimation	estimation	NOUN
ejpam-5009	37	32	of	of	ADP
ejpam-5009	37	33	the	the	DET
ejpam-5009	37	34	eigenvalues	eigenvalue	NOUN
ejpam-5009	37	35	is	be	AUX
ejpam-5009	37	36	presented	present	VERB
ejpam-5009	37	37	.	.	PUNCT
ejpam-5009	38	1	in	in	ADP
ejpam-5009	38	2	order	order	NOUN
ejpam-5009	38	3	to	to	PART
ejpam-5009	38	4	obtain	obtain	VERB
ejpam-5009	38	5	our	our	PRON
ejpam-5009	38	6	result	result	NOUN
ejpam-5009	38	7	,	,	PUNCT
ejpam-5009	38	8	we	we	PRON
ejpam-5009	38	9	need	need	VERB
ejpam-5009	38	10	the	the	DET
ejpam-5009	38	11	following	follow	VERB
ejpam-5009	38	12	lemmas	lemmas	NOUN
ejpam-5009	38	13	.	.	PUNCT
ejpam-5009	39	1	lemma	lemma	PROPN
ejpam-5009	39	2	1	1	NUM
ejpam-5009	39	3	.	.	PUNCT
ejpam-5009	40	1	[	[	X
ejpam-5009	40	2	7	7	X
ejpam-5009	40	3	]	]	PUNCT
ejpam-5009	40	4	let	let	VERB
ejpam-5009	40	5	a	a	DET
ejpam-5009	40	6	∈	∈	PROPN
ejpam-5009	40	7	mn	mn	PROPN
ejpam-5009	40	8	with	with	ADP
ejpam-5009	40	9	n	n	PROPN
ejpam-5009	40	10	≥	≥	NUM
ejpam-5009	40	11	3	3	NUM
ejpam-5009	40	12	,	,	PUNCT
ejpam-5009	40	13	then∣∣∣∣λl(a)−	then∣∣∣∣λl(a)−	PRON
ejpam-5009	40	14	tra	tra	NOUN
ejpam-5009	40	15	n	n	NOUN
ejpam-5009	40	16	∣∣∣∣2	∣∣∣∣2	NOUN
ejpam-5009	40	17	≤	≤	NUM
ejpam-5009	40	18	n−	n−	PROPN
ejpam-5009	40	19	1	1	NUM
ejpam-5009	40	20	n	n	CCONJ
ejpam-5009	40	21			PROPN
ejpam-5009	40	22	n∑	n∑	PROPN
ejpam-5009	40	23	j=1	j=1	ADJ
ejpam-5009	40	24	|λj(a)|2	|λj(a)|2	PROPN
ejpam-5009	40	25	−	−	ADV
ejpam-5009	40	26	|tra|2	|tra|2	ADV
ejpam-5009	41	1	n	n	CCONJ
ejpam-5009	41	2	−	−	PROPN
ejpam-5009	41	3	1	1	NUM
ejpam-5009	41	4	2	2	NUM
ejpam-5009	41	5	s2(a	s2(a	NOUN
ejpam-5009	41	6	)	)	PUNCT
ejpam-5009	41	7			PROPN
ejpam-5009	41	8	for	for	ADP
ejpam-5009	41	9	l	l	NOUN
ejpam-5009	41	10	=	=	SYM
ejpam-5009	41	11	1	1	NUM
ejpam-5009	41	12	,	,	PUNCT
ejpam-5009	41	13	2	2	NUM
ejpam-5009	41	14	,	,	PUNCT
ejpam-5009	41	15	·	·	PUNCT
ejpam-5009	41	16	·	·	PUNCT
ejpam-5009	41	17	·	·	PUNCT
ejpam-5009	41	18	,	,	PUNCT
ejpam-5009	41	19	n	n	PROPN
ejpam-5009	41	20	and	and	CCONJ
ejpam-5009	41	21	s(a	s(a	PROPN
ejpam-5009	41	22	)	)	PUNCT
ejpam-5009	42	1	=	=	SYM
ejpam-5009	42	2	min	min	PROPN
ejpam-5009	42	3	1≤l≤n	1≤l≤n	NUM
ejpam-5009	42	4	max	max	PROPN
ejpam-5009	42	5	j	j	PROPN
ejpam-5009	42	6	,	,	PUNCT
ejpam-5009	42	7	k∈sl	k∈sl	PROPN
ejpam-5009	42	8	|λj(a)−	|λj(a)−	NOUN
ejpam-5009	42	9	λk(a)|	λk(a)|	NUM
ejpam-5009	42	10	.	.	PUNCT
ejpam-5009	43	1	f.	f.	PROPN
ejpam-5009	43	2	zhang	zhang	PROPN
ejpam-5009	43	3	,	,	PUNCT
ejpam-5009	43	4	c.	c.	PROPN
ejpam-5009	43	5	zhang	zhang	PROPN
ejpam-5009	43	6	/	/	SYM
ejpam-5009	43	7	eur	eur	PROPN
ejpam-5009	43	8	.	.	PUNCT
ejpam-5009	44	1	j.	j.	PROPN
ejpam-5009	44	2	pure	pure	PROPN
ejpam-5009	44	3	appl	appl	PROPN
ejpam-5009	44	4	.	.	PROPN
ejpam-5009	44	5	math	math	PROPN
ejpam-5009	44	6	,	,	PUNCT
ejpam-5009	44	7	17	17	NUM
ejpam-5009	44	8	(	(	PUNCT
ejpam-5009	44	9	1	1	NUM
ejpam-5009	44	10	)	)	PUNCT
ejpam-5009	44	11	(	(	PUNCT
ejpam-5009	44	12	2024	2024	NUM
ejpam-5009	44	13	)	)	PUNCT
ejpam-5009	44	14	,	,	PUNCT
ejpam-5009	44	15	243	243	NUM
ejpam-5009	44	16	-	-	SYM
ejpam-5009	44	17	247	247	NUM
ejpam-5009	44	18	245	245	NUM
ejpam-5009	44	19	lemma	lemma	PROPN
ejpam-5009	44	20	2	2	NUM
ejpam-5009	44	21	.	.	PUNCT
ejpam-5009	45	1	[	[	X
ejpam-5009	45	2	7	7	X
ejpam-5009	45	3	]	]	PUNCT
ejpam-5009	45	4	let	let	VERB
ejpam-5009	45	5	a	a	DET
ejpam-5009	45	6	∈	∈	PROPN
ejpam-5009	45	7	mn	mn	PROPN
ejpam-5009	45	8	,	,	PUNCT
ejpam-5009	45	9	then	then	ADV
ejpam-5009	45	10	n∑	n∑	PROPN
ejpam-5009	45	11	j=1	j=1	PROPN
ejpam-5009	45	12	|λj(a)|2	|λj(a)|2	VERB
ejpam-5009	45	13	≤	≤	NUM
ejpam-5009	45	14	√√√√(∥a∥2f	√√√√(∥a∥2f	NUM
ejpam-5009	46	1	−	−	PROPN
ejpam-5009	46	2	|tra|2	|tra|2	PROPN
ejpam-5009	46	3	n	n	CCONJ
ejpam-5009	46	4	)	)	PUNCT
ejpam-5009	46	5	2	2	NUM
ejpam-5009	46	6	−	−	PROPN
ejpam-5009	46	7	∥[a	∥[a	PROPN
ejpam-5009	46	8	,	,	PUNCT
ejpam-5009	46	9	a∗]∥2f	a∗]∥2f	X
ejpam-5009	46	10	2	2	NUM
ejpam-5009	46	11	+	+	CCONJ
ejpam-5009	46	12	|tra|2	|tra|2	ADV
ejpam-5009	46	13	n	n	NOUN
ejpam-5009	46	14	.	.	PUNCT
ejpam-5009	47	1	next	next	ADV
ejpam-5009	47	2	we	we	PRON
ejpam-5009	47	3	give	give	VERB
ejpam-5009	47	4	a	a	DET
ejpam-5009	47	5	new	new	ADJ
ejpam-5009	47	6	proof	proof	NOUN
ejpam-5009	47	7	of	of	ADP
ejpam-5009	47	8	lemma	lemma	PROPN
ejpam-5009	47	9	2.2	2.2	NUM
ejpam-5009	47	10	in	in	ADP
ejpam-5009	47	11	[	[	X
ejpam-5009	47	12	6	6	NUM
ejpam-5009	47	13	]	]	PUNCT
ejpam-5009	47	14	,	,	PUNCT
ejpam-5009	47	15	which	which	PRON
ejpam-5009	47	16	plays	play	VERB
ejpam-5009	47	17	a	a	DET
ejpam-5009	47	18	key	key	ADJ
ejpam-5009	47	19	role	role	NOUN
ejpam-5009	47	20	in	in	ADP
ejpam-5009	47	21	their	their	PRON
ejpam-5009	47	22	discussion	discussion	NOUN
ejpam-5009	47	23	.	.	PUNCT
ejpam-5009	48	1	lemma	lemma	PROPN
ejpam-5009	48	2	3	3	X
ejpam-5009	48	3	.	.	PUNCT
ejpam-5009	49	1	let	let	VERB
ejpam-5009	49	2	m	m	VERB
ejpam-5009	49	3	=	=	PUNCT
ejpam-5009	49	4	[	[	PUNCT
ejpam-5009	49	5	ak	ak	PROPN
ejpam-5009	49	6	bk	bk	PROPN
ejpam-5009	49	7	,	,	PUNCT
ejpam-5009	49	8	n−k	n−k	NOUN
ejpam-5009	49	9	cn−k	cn−k	NOUN
ejpam-5009	49	10	,	,	PUNCT
ejpam-5009	49	11	k	k	PROPN
ejpam-5009	49	12	dn−k	dn−k	PROPN
ejpam-5009	49	13	]	]	PUNCT
ejpam-5009	49	14	with	with	ADP
ejpam-5009	49	15	eigenvalues	eigenvalue	NOUN
ejpam-5009	49	16	λ1	λ1	ADJ
ejpam-5009	49	17	,	,	PUNCT
ejpam-5009	49	18	λ2	λ2	NOUN
ejpam-5009	49	19	,	,	PUNCT
ejpam-5009	49	20	·	·	PUNCT
ejpam-5009	49	21	·	·	PUNCT
ejpam-5009	49	22	·	·	PUNCT
ejpam-5009	49	23	,	,	PUNCT
ejpam-5009	49	24	λn	λn	NOUN
ejpam-5009	49	25	,	,	PUNCT
ejpam-5009	49	26	then	then	ADV
ejpam-5009	49	27	n∑	n∑	PROPN
ejpam-5009	49	28	j=1	j=1	NOUN
ejpam-5009	49	29	|λj	|λj	NUM
ejpam-5009	49	30	|2	|2	NUM
ejpam-5009	49	31	≤	≤	NUM
ejpam-5009	49	32	min	min	NOUN
ejpam-5009	49	33	x	x	INTJ
ejpam-5009	49	34	̸=0	̸=0	ADJ
ejpam-5009	49	35	min	min	PROPN
ejpam-5009	49	36	1≤k≤n−1	1≤k≤n−1	NUM
ejpam-5009	49	37	fm	fm	PROPN
ejpam-5009	49	38	(	(	PUNCT
ejpam-5009	49	39	k	k	X
ejpam-5009	49	40	,	,	PUNCT
ejpam-5009	49	41	x	x	X
ejpam-5009	49	42	)	)	PUNCT
ejpam-5009	49	43	is	be	AUX
ejpam-5009	49	44	valid	valid	ADJ
ejpam-5009	49	45	for	for	ADP
ejpam-5009	49	46	any	any	DET
ejpam-5009	49	47	non	non	ADJ
ejpam-5009	49	48	-	-	ADJ
ejpam-5009	49	49	zero	zero	NUM
ejpam-5009	49	50	number	number	NOUN
ejpam-5009	49	51	x.	x.	NOUN
ejpam-5009	49	52	proof	proof	NOUN
ejpam-5009	49	53	.	.	PUNCT
ejpam-5009	50	1	let	let	VERB
ejpam-5009	50	2	x	x	PUNCT
ejpam-5009	50	3	=	=	PUNCT
ejpam-5009	51	1	[	[	PUNCT
ejpam-5009	51	2	xik	xik	NOUN
ejpam-5009	51	3	0	0	SYM
ejpam-5009	51	4	0	0	NUM
ejpam-5009	51	5	in−k	in−k	NOUN
ejpam-5009	51	6	]	]	PUNCT
ejpam-5009	51	7	,	,	PUNCT
ejpam-5009	51	8	then	then	ADV
ejpam-5009	51	9	m(x	m(x	NOUN
ejpam-5009	51	10	)	)	PUNCT
ejpam-5009	52	1	=	=	SYM
ejpam-5009	52	2	xmx−1	xmx−1	PROPN
ejpam-5009	52	3	,	,	PUNCT
ejpam-5009	52	4	where	where	SCONJ
ejpam-5009	52	5	ik	ik	PROPN
ejpam-5009	52	6	is	be	AUX
ejpam-5009	52	7	a	a	DET
ejpam-5009	52	8	k	k	PROPN
ejpam-5009	52	9	×	×	PROPN
ejpam-5009	52	10	k	k	PROPN
ejpam-5009	52	11	unit	unit	NOUN
ejpam-5009	52	12	matrix	matrix	NOUN
ejpam-5009	52	13	.	.	PUNCT
ejpam-5009	53	1	obviously	obviously	ADV
ejpam-5009	53	2	,	,	PUNCT
ejpam-5009	53	3	m(x	m(x	PROPN
ejpam-5009	53	4	)	)	PUNCT
ejpam-5009	53	5	is	be	AUX
ejpam-5009	53	6	similar	similar	ADJ
ejpam-5009	53	7	to	to	ADP
ejpam-5009	53	8	m	m	PRON
ejpam-5009	53	9	.	.	PUNCT
ejpam-5009	54	1	by	by	ADP
ejpam-5009	54	2	lemma	lemma	PROPN
ejpam-5009	54	3	[	[	X
ejpam-5009	54	4	2	2	NUM
ejpam-5009	54	5	]	]	PUNCT
ejpam-5009	54	6	,	,	PUNCT
ejpam-5009	54	7	we	we	PRON
ejpam-5009	54	8	have	have	VERB
ejpam-5009	54	9	n∑	n∑	ADJ
ejpam-5009	54	10	j=1	j=1	ADJ
ejpam-5009	54	11	|λj(m)|2	|λj(m)|2	NOUN
ejpam-5009	54	12	=	=	SYM
ejpam-5009	55	1	n∑	n∑	PROPN
ejpam-5009	55	2	j=1	j=1	PROPN
ejpam-5009	55	3	|λj(m(x))|2	|λj(m(x))|2	PROPN
ejpam-5009	55	4	(	(	PUNCT
ejpam-5009	55	5	4	4	X
ejpam-5009	55	6	)	)	PUNCT
ejpam-5009	55	7	≤	≤	NOUN
ejpam-5009	55	8	√√√√(∥m(x)∥2f	√√√√(∥m(x)∥2f	NOUN
ejpam-5009	55	9	−	−	NOUN
ejpam-5009	55	10	|trm(x)|2	|trm(x)|2	PROPN
ejpam-5009	55	11	n	n	X
ejpam-5009	55	12	)	)	PUNCT
ejpam-5009	55	13	2	2	NUM
ejpam-5009	55	14	−	−	NOUN
ejpam-5009	55	15	∥[m(x),m(x)∗]∥2f	∥[m(x),m(x)∗]∥2f	NOUN
ejpam-5009	55	16	2	2	NUM
ejpam-5009	55	17	+	+	NOUN
ejpam-5009	55	18	|trm(x)|2	|trm(x)|2	NOUN
ejpam-5009	55	19	n	n	NOUN
ejpam-5009	55	20	=	=	VERB
ejpam-5009	55	21	√√√√(∥m(x)∥2f	√√√√(∥m(x)∥2f	NOUN
ejpam-5009	55	22	−	−	PROPN
ejpam-5009	55	23	|trm	|trm	ADJ
ejpam-5009	55	24	|2	|2	NUM
ejpam-5009	55	25	n	n	CCONJ
ejpam-5009	55	26	)	)	SYM
ejpam-5009	55	27	2	2	NUM
ejpam-5009	55	28	−	−	NOUN
ejpam-5009	55	29	∥[m(x),m(x)∗]∥2f	∥[m(x),m(x)∗]∥2f	NOUN
ejpam-5009	55	30	2	2	NUM
ejpam-5009	55	31	+	+	NUM
ejpam-5009	55	32	|trm	|trm	ADJ
ejpam-5009	55	33	|2	|2	NUM
ejpam-5009	55	34	n	n	NOUN
ejpam-5009	55	35	,	,	PUNCT
ejpam-5009	55	36	where	where	SCONJ
ejpam-5009	55	37	∥m(x)∥f	∥m(x)∥f	PROPN
ejpam-5009	55	38	=	=	SYM
ejpam-5009	55	39	(	(	PUNCT
ejpam-5009	55	40	∥m∥2f	∥m∥2f	NUM
ejpam-5009	55	41	−	−	PUNCT
ejpam-5009	56	1	[	[	X
ejpam-5009	56	2	(	(	PUNCT
ejpam-5009	56	3	1−	1−	NUM
ejpam-5009	56	4	x2	x2	PROPN
ejpam-5009	56	5	)	)	PUNCT
ejpam-5009	56	6	∥bk	∥bk	ADP
ejpam-5009	56	7	,	,	PUNCT
ejpam-5009	56	8	n−k∥2f	n−k∥2f	NOUN
ejpam-5009	56	9	+	+	CCONJ
ejpam-5009	56	10	(	(	PUNCT
ejpam-5009	56	11	1−	1−	NUM
ejpam-5009	56	12	x−2	x−2	PROPN
ejpam-5009	56	13	)	)	PUNCT
ejpam-5009	56	14	∥cn−k	∥cn−k	PROPN
ejpam-5009	56	15	,	,	PUNCT
ejpam-5009	56	16	k∥2f	k∥2f	NOUN
ejpam-5009	56	17	]	]	PUNCT
ejpam-5009	56	18	)	)	PUNCT
ejpam-5009	56	19	1	1	NUM
ejpam-5009	56	20	2	2	NUM
ejpam-5009	56	21	.	.	PUNCT
ejpam-5009	57	1	(	(	PUNCT
ejpam-5009	57	2	5	5	X
ejpam-5009	57	3	)	)	PUNCT
ejpam-5009	57	4	combing	comb	VERB
ejpam-5009	57	5	inequality	inequality	NOUN
ejpam-5009	57	6	(	(	PUNCT
ejpam-5009	57	7	4	4	NUM
ejpam-5009	57	8	)	)	PUNCT
ejpam-5009	57	9	and	and	CCONJ
ejpam-5009	57	10	equality	equality	NOUN
ejpam-5009	57	11	(	(	PUNCT
ejpam-5009	57	12	5	5	NUM
ejpam-5009	57	13	)	)	PUNCT
ejpam-5009	57	14	,	,	PUNCT
ejpam-5009	57	15	we	we	PRON
ejpam-5009	57	16	conclude	conclude	VERB
ejpam-5009	57	17	lemma	lemma	PROPN
ejpam-5009	57	18	3	3	X
ejpam-5009	57	19	.	.	PUNCT
ejpam-5009	58	1	we	we	PRON
ejpam-5009	58	2	now	now	ADV
ejpam-5009	58	3	focus	focus	VERB
ejpam-5009	58	4	on	on	ADP
ejpam-5009	58	5	the	the	DET
ejpam-5009	58	6	location	location	NOUN
ejpam-5009	58	7	of	of	ADP
ejpam-5009	58	8	the	the	DET
ejpam-5009	58	9	eigenvalues	eigenvalue	NOUN
ejpam-5009	58	10	of	of	ADP
ejpam-5009	58	11	complex	complex	ADJ
ejpam-5009	58	12	matrices	matrix	NOUN
ejpam-5009	58	13	.	.	PUNCT
ejpam-5009	59	1	theorem	theorem	NOUN
ejpam-5009	59	2	1	1	NUM
ejpam-5009	59	3	.	.	PUNCT
ejpam-5009	60	1	let	let	VERB
ejpam-5009	60	2	m	m	VERB
ejpam-5009	60	3	=	=	PUNCT
ejpam-5009	60	4	[	[	PUNCT
ejpam-5009	60	5	ak	ak	PROPN
ejpam-5009	60	6	bk	bk	PROPN
ejpam-5009	60	7	,	,	PUNCT
ejpam-5009	60	8	n−k	n−k	NOUN
ejpam-5009	60	9	cn−k	cn−k	NOUN
ejpam-5009	60	10	,	,	PUNCT
ejpam-5009	60	11	k	k	PROPN
ejpam-5009	60	12	dn−k	dn−k	PROPN
ejpam-5009	60	13	]	]	PUNCT
ejpam-5009	60	14	with	with	ADP
ejpam-5009	60	15	eigenvalues	eigenvalue	NOUN
ejpam-5009	60	16	λ1	λ1	ADJ
ejpam-5009	60	17	,	,	PUNCT
ejpam-5009	60	18	λ2	λ2	NOUN
ejpam-5009	60	19	,	,	PUNCT
ejpam-5009	60	20	·	·	PUNCT
ejpam-5009	60	21	·	·	PUNCT
ejpam-5009	60	22	·	·	PUNCT
ejpam-5009	60	23	,	,	PUNCT
ejpam-5009	60	24	λn	λn	X
ejpam-5009	60	25	(	(	PUNCT
ejpam-5009	60	26	n	n	CCONJ
ejpam-5009	60	27	≥	≥	NOUN
ejpam-5009	60	28	3	3	NUM
ejpam-5009	60	29	)	)	PUNCT
ejpam-5009	60	30	,	,	PUNCT
ejpam-5009	60	31	then	then	ADV
ejpam-5009	60	32	all	all	PRON
ejpam-5009	60	33	of	of	ADP
ejpam-5009	60	34	eigenvalues	eigenvalue	NOUN
ejpam-5009	60	35	of	of	ADP
ejpam-5009	60	36	m	m	NOUN
ejpam-5009	60	37	are	be	AUX
ejpam-5009	60	38	included	include	VERB
ejpam-5009	60	39	by	by	ADP
ejpam-5009	60	40	the	the	DET
ejpam-5009	60	41	following	follow	VERB
ejpam-5009	60	42	disk	disk	NOUN
ejpam-5009	60	43	:	:	PUNCT
ejpam-5009	61	1	∣∣∣∣λl(m)−	∣∣∣∣λl(m)−	PROPN
ejpam-5009	61	2	trm	trm	ADJ
ejpam-5009	61	3	n	n	CCONJ
ejpam-5009	61	4	∣∣∣∣	∣∣∣∣	NOUN
ejpam-5009	61	5	≤	≤	NUM
ejpam-5009	61	6	min	min	NOUN
ejpam-5009	62	1	x	x	INTJ
ejpam-5009	62	2	̸=0	̸=0	ADJ
ejpam-5009	62	3	min	min	PROPN
ejpam-5009	62	4	1≤k≤n−1	1≤k≤n−1	NUM
ejpam-5009	62	5	√	√	NUM
ejpam-5009	62	6	n−	n−	NOUN
ejpam-5009	62	7	1	1	NUM
ejpam-5009	62	8	n	n	PROPN
ejpam-5009	62	9	(	(	PUNCT
ejpam-5009	62	10	fm	fm	PROPN
ejpam-5009	62	11	(	(	PUNCT
ejpam-5009	62	12	k	k	X
ejpam-5009	62	13	,	,	PUNCT
ejpam-5009	62	14	x)−	x)−	PROPN
ejpam-5009	62	15	|trm	|trm	ADJ
ejpam-5009	62	16	|2	|2	NUM
ejpam-5009	62	17	n	n	CCONJ
ejpam-5009	62	18	−	−	NUM
ejpam-5009	62	19	1	1	NUM
ejpam-5009	62	20	2	2	NUM
ejpam-5009	62	21	s2(m	s2(m	NOUN
ejpam-5009	62	22	)	)	PUNCT
ejpam-5009	62	23	)	)	PUNCT
ejpam-5009	62	24	1	1	NUM
ejpam-5009	62	25	2	2	NUM
ejpam-5009	62	26	for	for	ADP
ejpam-5009	62	27	l	l	NOUN
ejpam-5009	62	28	=	=	SYM
ejpam-5009	62	29	1	1	NUM
ejpam-5009	62	30	,	,	PUNCT
ejpam-5009	62	31	2	2	NUM
ejpam-5009	62	32	,	,	PUNCT
ejpam-5009	62	33	·	·	PUNCT
ejpam-5009	62	34	·	·	PUNCT
ejpam-5009	62	35	·	·	PUNCT
ejpam-5009	62	36	,	,	PUNCT
ejpam-5009	62	37	n	n	PROPN
ejpam-5009	62	38	and	and	CCONJ
ejpam-5009	62	39	s(m	s(m	PROPN
ejpam-5009	62	40	)	)	PUNCT
ejpam-5009	63	1	=	=	SYM
ejpam-5009	63	2	min	min	PROPN
ejpam-5009	63	3	1≤l≤n	1≤l≤n	NUM
ejpam-5009	63	4	max	max	PROPN
ejpam-5009	63	5	j	j	PROPN
ejpam-5009	63	6	,	,	PUNCT
ejpam-5009	63	7	k∈sl	k∈sl	PROPN
ejpam-5009	63	8	|λj(m)−	|λj(m)−	PROPN
ejpam-5009	63	9	λk(m)|	λk(m)|	NOUN
ejpam-5009	63	10	.	.	PUNCT
ejpam-5009	63	11	references	reference	NOUN
ejpam-5009	63	12	246	246	NUM
ejpam-5009	63	13	proof	proof	NOUN
ejpam-5009	63	14	.	.	PUNCT
ejpam-5009	64	1	combining	combine	VERB
ejpam-5009	64	2	lemmas	lemmas	PROPN
ejpam-5009	64	3	2.1	2.1	NUM
ejpam-5009	64	4	and	and	CCONJ
ejpam-5009	64	5	2.3	2.3	NUM
ejpam-5009	64	6	,	,	PUNCT
ejpam-5009	64	7	we	we	PRON
ejpam-5009	64	8	deduce	deduce	VERB
ejpam-5009	64	9	that∣∣∣∣λl(m)−	that∣∣∣∣λl(m)−	NOUN
ejpam-5009	64	10	trm	trm	PROPN
ejpam-5009	64	11	n	n	CCONJ
ejpam-5009	64	12	∣∣∣∣2	∣∣∣∣2	NOUN
ejpam-5009	64	13	≤	≤	NUM
ejpam-5009	64	14	n−	n−	PROPN
ejpam-5009	64	15	1	1	NUM
ejpam-5009	64	16	n	n	CCONJ
ejpam-5009	64	17			PROPN
ejpam-5009	64	18	n∑	n∑	PROPN
ejpam-5009	64	19	j=1	j=1	PROPN
ejpam-5009	64	20	|λj(m)|2	|λj(m)|2	PROPN
ejpam-5009	64	21	−	−	PROPN
ejpam-5009	64	22	|trm	|trm	ADJ
ejpam-5009	64	23	|2	|2	NUM
ejpam-5009	64	24	n	n	CCONJ
ejpam-5009	64	25	−	−	NUM
ejpam-5009	64	26	1	1	NUM
ejpam-5009	64	27	2	2	NUM
ejpam-5009	64	28	s2(m	s2(m	NOUN
ejpam-5009	64	29	)	)	PUNCT
ejpam-5009	64	30			PROPN
ejpam-5009	64	31	≤	≤	ADJ
ejpam-5009	64	32	n−	n−	NOUN
ejpam-5009	64	33	1	1	NUM
ejpam-5009	64	34	n	n	PROPN
ejpam-5009	64	35	(	(	PUNCT
ejpam-5009	64	36	min	min	NOUN
ejpam-5009	64	37	x	x	INTJ
ejpam-5009	64	38	̸=0	̸=0	ADJ
ejpam-5009	64	39	min	min	PROPN
ejpam-5009	64	40	1≤k≤n−1	1≤k≤n−1	NUM
ejpam-5009	64	41	fm	fm	PROPN
ejpam-5009	64	42	(	(	PUNCT
ejpam-5009	64	43	k	k	X
ejpam-5009	64	44	,	,	PUNCT
ejpam-5009	64	45	x)−	x)−	PROPN
ejpam-5009	64	46	|trm	|trm	ADJ
ejpam-5009	64	47	|2	|2	NUM
ejpam-5009	64	48	n	n	CCONJ
ejpam-5009	64	49	−	−	NUM
ejpam-5009	64	50	1	1	NUM
ejpam-5009	64	51	2	2	NUM
ejpam-5009	64	52	s2(m	s2(m	NOUN
ejpam-5009	64	53	)	)	PUNCT
ejpam-5009	64	54	)	)	PUNCT
ejpam-5009	65	1	≤	≤	NUM
ejpam-5009	65	2	min	min	NOUN
ejpam-5009	65	3	x̸=0	x̸=0	PROPN
ejpam-5009	65	4	min	min	PROPN
ejpam-5009	65	5	1≤k≤n−1	1≤k≤n−1	NUM
ejpam-5009	65	6	n−	n−	PROPN
ejpam-5009	65	7	1	1	NUM
ejpam-5009	65	8	n	n	PROPN
ejpam-5009	65	9	(	(	PUNCT
ejpam-5009	65	10	fm	fm	PROPN
ejpam-5009	65	11	(	(	PUNCT
ejpam-5009	65	12	k	k	X
ejpam-5009	65	13	,	,	PUNCT
ejpam-5009	65	14	x)−	x)−	PROPN
ejpam-5009	65	15	|trm	|trm	ADJ
ejpam-5009	65	16	|2	|2	NUM
ejpam-5009	65	17	n	n	CCONJ
ejpam-5009	65	18	−	−	NUM
ejpam-5009	65	19	1	1	NUM
ejpam-5009	65	20	2	2	NUM
ejpam-5009	65	21	s2(m	s2(m	NOUN
ejpam-5009	65	22	)	)	PUNCT
ejpam-5009	65	23	)	)	PUNCT
ejpam-5009	65	24	.	.	PUNCT
ejpam-5009	66	1	therefore,∣∣∣∣λl(m)−	therefore,∣∣∣∣λl(m)−	PROPN
ejpam-5009	66	2	trm	trm	PROPN
ejpam-5009	66	3	n	n	CCONJ
ejpam-5009	66	4	∣∣∣∣	∣∣∣∣	NOUN
ejpam-5009	66	5	≤	≤	NUM
ejpam-5009	66	6	min	min	NOUN
ejpam-5009	66	7	x	x	INTJ
ejpam-5009	66	8	̸=0	̸=0	ADJ
ejpam-5009	66	9	min	min	PROPN
ejpam-5009	66	10	1≤k≤n−1	1≤k≤n−1	NUM
ejpam-5009	66	11	√	√	NUM
ejpam-5009	66	12	n−	n−	NOUN
ejpam-5009	66	13	1	1	NUM
ejpam-5009	66	14	n	n	PROPN
ejpam-5009	66	15	(	(	PUNCT
ejpam-5009	66	16	fm	fm	PROPN
ejpam-5009	66	17	(	(	PUNCT
ejpam-5009	66	18	k	k	X
ejpam-5009	66	19	,	,	PUNCT
ejpam-5009	66	20	x)−	x)−	PROPN
ejpam-5009	66	21	|trm	|trm	ADJ
ejpam-5009	66	22	|2	|2	NUM
ejpam-5009	66	23	n	n	CCONJ
ejpam-5009	66	24	−	−	NUM
ejpam-5009	66	25	1	1	NUM
ejpam-5009	66	26	2	2	NUM
ejpam-5009	66	27	s2(m	s2(m	NOUN
ejpam-5009	66	28	)	)	PUNCT
ejpam-5009	66	29	)	)	PUNCT
ejpam-5009	66	30	1	1	NUM
ejpam-5009	66	31	2	2	NUM
ejpam-5009	66	32	for	for	ADP
ejpam-5009	66	33	s(m	s(m	NOUN
ejpam-5009	66	34	)	)	PUNCT
ejpam-5009	67	1	=	=	SYM
ejpam-5009	67	2	min	min	PROPN
ejpam-5009	67	3	1≤l≤n	1≤l≤n	NUM
ejpam-5009	67	4	max	max	PROPN
ejpam-5009	67	5	j	j	PROPN
ejpam-5009	67	6	,	,	PUNCT
ejpam-5009	67	7	k∈sl	k∈sl	PROPN
ejpam-5009	67	8	|λj(m)−	|λj(m)−	PROPN
ejpam-5009	67	9	λk(m)|	λk(m)|	NOUN
ejpam-5009	67	10	.	.	PUNCT
ejpam-5009	68	1	this	this	PRON
ejpam-5009	68	2	completed	complete	VERB
ejpam-5009	68	3	the	the	DET
ejpam-5009	68	4	proof	proof	NOUN
ejpam-5009	68	5	.	.	PUNCT
ejpam-5009	69	1	for	for	ADP
ejpam-5009	69	2	complex	complex	ADJ
ejpam-5009	69	3	matrix	matrix	NOUN
ejpam-5009	69	4	with	with	ADP
ejpam-5009	69	5	order	order	NOUN
ejpam-5009	69	6	n(n	n(n	PROPN
ejpam-5009	69	7	>	>	X
ejpam-5009	69	8	2	2	NUM
ejpam-5009	69	9	)	)	PUNCT
ejpam-5009	69	10	,	,	PUNCT
ejpam-5009	69	11	then	then	ADV
ejpam-5009	69	12	the	the	DET
ejpam-5009	69	13	computation	computation	NOUN
ejpam-5009	69	14	of	of	ADP
ejpam-5009	69	15	theorem	theorem	ADJ
ejpam-5009	69	16	2.4	2.4	NUM
ejpam-5009	69	17	requires	require	VERB
ejpam-5009	69	18	approximately	approximately	ADV
ejpam-5009	69	19	n3	n3	ADJ
ejpam-5009	69	20	2	2	NUM
ejpam-5009	69	21	additional	additional	ADJ
ejpam-5009	69	22	calculations	calculation	NOUN
ejpam-5009	69	23	compared	compare	VERB
ejpam-5009	69	24	to	to	ADP
ejpam-5009	69	25	the	the	DET
ejpam-5009	69	26	computation	computation	NOUN
ejpam-5009	69	27	of	of	ADP
ejpam-5009	69	28	inequality(3	inequality(3	NOUN
ejpam-5009	69	29	)	)	PUNCT
ejpam-5009	69	30	.	.	PUNCT
ejpam-5009	70	1	this	this	PRON
ejpam-5009	70	2	indicates	indicate	VERB
ejpam-5009	70	3	that	that	SCONJ
ejpam-5009	70	4	its	its	PRON
ejpam-5009	70	5	computational	computational	ADJ
ejpam-5009	70	6	complexity	complexity	NOUN
ejpam-5009	70	7	is	be	AUX
ejpam-5009	70	8	greater	great	ADJ
ejpam-5009	70	9	than	than	ADP
ejpam-5009	70	10	the	the	DET
ejpam-5009	70	11	computational	computational	ADJ
ejpam-5009	70	12	complexity	complexity	NOUN
ejpam-5009	70	13	in	in	ADP
ejpam-5009	70	14	(	(	PUNCT
ejpam-5009	70	15	3	3	NUM
ejpam-5009	70	16	)	)	PUNCT
ejpam-5009	70	17	.	.	PUNCT
ejpam-5009	71	1	but	but	CCONJ
ejpam-5009	71	2	,	,	PUNCT
ejpam-5009	71	3	in	in	ADP
ejpam-5009	71	4	theory	theory	NOUN
ejpam-5009	71	5	,	,	PUNCT
ejpam-5009	71	6	theorem	theorem	VERB
ejpam-5009	71	7	2.4	2.4	NUM
ejpam-5009	71	8	is	be	AUX
ejpam-5009	71	9	a	a	DET
ejpam-5009	71	10	refinement	refinement	NOUN
ejpam-5009	71	11	of	of	ADP
ejpam-5009	71	12	(	(	PUNCT
ejpam-5009	71	13	3	3	NUM
ejpam-5009	71	14	)	)	PUNCT
ejpam-5009	71	15	.	.	PUNCT
ejpam-5009	72	1	3	3	X
ejpam-5009	72	2	.	.	X
ejpam-5009	72	3	funding	fund	VERB
ejpam-5009	72	4	feng	feng	PROPN
ejpam-5009	72	5	zhang	zhang	PROPN
ejpam-5009	72	6	is	be	AUX
ejpam-5009	72	7	supported	support	VERB
ejpam-5009	72	8	by	by	ADP
ejpam-5009	72	9	the	the	DET
ejpam-5009	72	10	southeast	southeast	PROPN
ejpam-5009	72	11	university	university	PROPN
ejpam-5009	72	12	chengxian	chengxian	PROPN
ejpam-5009	72	13	college	college	PROPN
ejpam-5009	72	14	young	young	ADJ
ejpam-5009	72	15	teacher	teacher	NOUN
ejpam-5009	72	16	research	research	PROPN
ejpam-5009	72	17	development	development	PROPN
ejpam-5009	72	18	fund	fund	PROPN
ejpam-5009	72	19	projection	projection	NOUN
ejpam-5009	72	20	(	(	PUNCT
ejpam-5009	72	21	no	no	INTJ
ejpam-5009	72	22	.	.	PUNCT
ejpam-5009	72	23	z0058	z0058	NUM
ejpam-5009	72	24	)	)	PUNCT
ejpam-5009	72	25	.	.	PUNCT
ejpam-5009	73	1	acknowledgements	acknowledgement	NOUN
ejpam-5009	73	2	feng	feng	PROPN
ejpam-5009	73	3	zhang	zhang	PROPN
ejpam-5009	73	4	thanks	thanks	PROPN
ejpam-5009	73	5	shiyu	shiyu	PROPN
ejpam-5009	73	6	xu	xu	PROPN
ejpam-5009	73	7	for	for	ADP
ejpam-5009	73	8	her	her	PRON
ejpam-5009	73	9	kindly	kindly	ADJ
ejpam-5009	73	10	help	help	NOUN
ejpam-5009	73	11	and	and	CCONJ
ejpam-5009	73	12	love	love	VERB
ejpam-5009	73	13	her	she	PRON
ejpam-5009	73	14	forever	forever	ADV
ejpam-5009	73	15	.	.	PUNCT
ejpam-5009	74	1	references	reference	NOUN
ejpam-5009	74	2	references	reference	NOUN
ejpam-5009	74	3	[	[	X
ejpam-5009	74	4	1	1	NUM
ejpam-5009	74	5	]	]	PUNCT
ejpam-5009	74	6	r.	r.	PROPN
ejpam-5009	74	7	a.	a.	PROPN
ejpam-5009	74	8	horn	horn	PROPN
ejpam-5009	74	9	and	and	CCONJ
ejpam-5009	74	10	c.	c.	PROPN
ejpam-5009	74	11	r.	r.	PROPN
ejpam-5009	74	12	johnson	johnson	PROPN
ejpam-5009	74	13	,	,	PUNCT
ejpam-5009	74	14	matrix	matrix	VERB
ejpam-5009	74	15	analysis	analysis	NOUN
ejpam-5009	74	16	.	.	PUNCT
ejpam-5009	75	1	1985	1985	NUM
ejpam-5009	75	2	.	.	PUNCT
ejpam-5009	76	1	[	[	X
ejpam-5009	76	2	2	2	NUM
ejpam-5009	76	3	]	]	PUNCT
ejpam-5009	76	4	r.	r.	PROPN
ejpam-5009	76	5	bhatia	bhatia	PROPN
ejpam-5009	76	6	,	,	PUNCT
ejpam-5009	76	7	positive	positive	ADJ
ejpam-5009	76	8	definite	definite	ADJ
ejpam-5009	76	9	matrices.princeton	matrices.princeton	PROPN
ejpam-5009	76	10	university	university	NOUN
ejpam-5009	76	11	press	press	NOUN
ejpam-5009	76	12	,	,	PUNCT
ejpam-5009	76	13	2007	2007	NUM
ejpam-5009	76	14	.	.	PUNCT
ejpam-5009	77	1	[	[	X
ejpam-5009	77	2	3	3	X
ejpam-5009	77	3	]	]	X
ejpam-5009	77	4	b.	b.	PROPN
ejpam-5009	77	5	tu	tu	PROPN
ejpam-5009	77	6	.	.	PUNCT
ejpam-5009	78	1	the	the	DET
ejpam-5009	78	2	lower	lower	ADV
ejpam-5009	78	3	bound	bind	VERB
ejpam-5009	78	4	of	of	ADP
ejpam-5009	78	5	the	the	DET
ejpam-5009	78	6	rank	rank	NOUN
ejpam-5009	78	7	and	and	CCONJ
ejpam-5009	78	8	non	non	ADJ
ejpam-5009	78	9	-	-	NOUN
ejpam-5009	78	10	singularity	singularity	NOUN
ejpam-5009	78	11	of	of	ADP
ejpam-5009	78	12	matrices	matrix	NOUN
ejpam-5009	78	13	.	.	PUNCT
ejpam-5009	79	1	coll	coll	PROPN
ejpam-5009	79	2	.	.	PUNCT
ejpam-5009	80	1	j.	j.	PROPN
ejpam-5009	80	2	fu	fu	PROPN
ejpam-5009	80	3	dan	dan	PROPN
ejpam-5009	80	4	univ	univ	PROPN
ejpam-5009	80	5	.	.	PROPN
ejpam-5009	80	6	,	,	PUNCT
ejpam-5009	81	1	416–422	416–422	NUM
ejpam-5009	81	2	,	,	PUNCT
ejpam-5009	81	3	1982	1982	NUM
ejpam-5009	81	4	.	.	PUNCT
ejpam-5009	82	1	[	[	X
ejpam-5009	82	2	4	4	NUM
ejpam-5009	82	3	]	]	X
ejpam-5009	82	4	y.	y.	PROPN
ejpam-5009	82	5	gu	gu	PROPN
ejpam-5009	82	6	.	.	PUNCT
ejpam-5009	83	1	the	the	DET
ejpam-5009	83	2	distribution	distribution	NOUN
ejpam-5009	83	3	of	of	ADP
ejpam-5009	83	4	eigenvalues	eigenvalue	NOUN
ejpam-5009	83	5	of	of	ADP
ejpam-5009	83	6	a	a	DET
ejpam-5009	83	7	matrix	matrix	NOUN
ejpam-5009	83	8	,	,	PUNCT
ejpam-5009	83	9	acta	acta	PROPN
ejpam-5009	83	10	math	math	PROPN
ejpam-5009	83	11	.	.	PUNCT
ejpam-5009	84	1	appl	appl	PROPN
ejpam-5009	84	2	.	.	PROPN
ejpam-5009	85	1	sinica	sinica	PROPN
ejpam-5009	85	2	,	,	PUNCT
ejpam-5009	85	3	501–511	501–511	NUM
ejpam-5009	85	4	,	,	PUNCT
ejpam-5009	85	5	1994	1994	NUM
ejpam-5009	85	6	.	.	PUNCT
ejpam-5009	86	1	references	reference	NOUN
ejpam-5009	86	2	247	247	NUM
ejpam-5009	87	1	[	[	SYM
ejpam-5009	87	2	5	5	NUM
ejpam-5009	87	3	]	]	PUNCT
ejpam-5009	87	4	l.	l.	PROPN
ejpam-5009	87	5	zou	zou	PROPN
ejpam-5009	87	6	,	,	PUNCT
ejpam-5009	87	7	y.	y.	PROPN
ejpam-5009	87	8	jiang	jiang	PROPN
ejpam-5009	87	9	.	.	PUNCT
ejpam-5009	88	1	estimation	estimation	NOUN
ejpam-5009	88	2	of	of	ADP
ejpam-5009	88	3	the	the	DET
ejpam-5009	88	4	eigenvalues	eigenvalue	NOUN
ejpam-5009	88	5	and	and	CCONJ
ejpam-5009	88	6	the	the	DET
ejpam-5009	88	7	smallest	small	ADJ
ejpam-5009	88	8	singular	singular	ADJ
ejpam-5009	88	9	value	value	NOUN
ejpam-5009	88	10	of	of	ADP
ejpam-5009	88	11	matrices	matrix	NOUN
ejpam-5009	88	12	,	,	PUNCT
ejpam-5009	88	13	linear	linear	ADJ
ejpam-5009	88	14	algebra	algebra	PROPN
ejpam-5009	88	15	appl	appl	NOUN
ejpam-5009	88	16	.	.	PROPN
ejpam-5009	88	17	,	,	PUNCT
ejpam-5009	88	18	1203–1211	1203–1211	NUM
ejpam-5009	88	19	,	,	PUNCT
ejpam-5009	88	20	2010	2010	NUM
ejpam-5009	88	21	.	.	PUNCT
ejpam-5009	89	1	[	[	X
ejpam-5009	89	2	6	6	NUM
ejpam-5009	89	3	]	]	PUNCT
ejpam-5009	89	4	j.	j.	PROPN
ejpam-5009	89	5	wu	wu	PROPN
ejpam-5009	89	6	,	,	PUNCT
ejpam-5009	89	7	j.	j.	PROPN
ejpam-5009	89	8	zhao	zhao	PROPN
ejpam-5009	89	9	,	,	PUNCT
ejpam-5009	89	10	a	a	DET
ejpam-5009	89	11	survey	survey	NOUN
ejpam-5009	89	12	of	of	ADP
ejpam-5009	89	13	the	the	DET
ejpam-5009	89	14	progress	progress	NOUN
ejpam-5009	89	15	of	of	ADP
ejpam-5009	89	16	locating	locate	VERB
ejpam-5009	89	17	methods	method	NOUN
ejpam-5009	89	18	of	of	ADP
ejpam-5009	89	19	complex	complex	ADJ
ejpam-5009	89	20	matrices	matrix	NOUN
ejpam-5009	89	21	eigenvalues	eigenvalue	VERB
ejpam-5009	89	22	and	and	CCONJ
ejpam-5009	89	23	some	some	DET
ejpam-5009	89	24	new	new	ADJ
ejpam-5009	89	25	location	location	NOUN
ejpam-5009	89	26	theorem	theorem	NOUN
ejpam-5009	89	27	and	and	CCONJ
ejpam-5009	89	28	their	their	PRON
ejpam-5009	89	29	applications	application	NOUN
ejpam-5009	89	30	,	,	PUNCT
ejpam-5009	89	31	i	i	PRON
ejpam-5009	89	32	m	m	VERB
ejpam-5009	89	33	a	a	PROPN
ejpam-5009	89	34	j.	j.	PROPN
ejpam-5009	89	35	appl	appl	PROPN
ejpam-5009	89	36	.	.	PROPN
ejpam-5009	89	37	math	math	PROPN
ejpam-5009	89	38	.	.	PUNCT
ejpam-5009	89	39	,	,	PUNCT
ejpam-5009	89	40	273–285	273–285	NUM
ejpam-5009	89	41	,	,	PUNCT
ejpam-5009	89	42	2015	2015	NUM
ejpam-5009	89	43	.	.	PUNCT
ejpam-5009	90	1	[	[	X
ejpam-5009	90	2	7	7	X
ejpam-5009	90	3	]	]	X
ejpam-5009	90	4	i.	i.	PROPN
ejpam-5009	90	5	h.	h.	PROPN
ejpam-5009	90	6	gumus	gumus	PROPN
ejpam-5009	90	7	,	,	PUNCT
ejpam-5009	90	8	o.	o.	PROPN
ejpam-5009	90	9	hirzallah	hirzallah	NOUN
ejpam-5009	90	10	,	,	PUNCT
ejpam-5009	90	11	and	and	CCONJ
ejpam-5009	90	12	f.	f.	PROPN
ejpam-5009	90	13	kittaneh	kittaneh	PROPN
ejpam-5009	90	14	,	,	PUNCT
ejpam-5009	90	15	eigenvalue	eigenvalue	ADJ
ejpam-5009	90	16	localization	localization	NOUN
ejpam-5009	90	17	for	for	ADP
ejpam-5009	90	18	complex	complex	ADJ
ejpam-5009	90	19	matrices	matrix	NOUN
ejpam-5009	90	20	,	,	PUNCT
ejpam-5009	90	21	electronic	electronic	ADJ
ejpam-5009	90	22	journal	journal	NOUN
ejpam-5009	90	23	of	of	ADP
ejpam-5009	90	24	linear	linear	PROPN
ejpam-5009	90	25	algebra	algebra	PROPN
ejpam-5009	90	26	,	,	PUNCT
ejpam-5009	90	27	892–906	892–906	NUM
ejpam-5009	90	28	,	,	PUNCT
ejpam-5009	90	29	2014	2014	NUM
ejpam-5009	90	30	.	.	PUNCT