CUJlIOO Mal.em4tico Educacional 
Vol. 5, N2 t, (W-tt9). J UNIO eoos. 

T hree proofs of an ident ity involving deri vati ves of a 
positive definite matrix and its determinant 

Ba i- Ni G uo • 
Depart ment of Appl ied Mathematics and Informatics 

Jia ozuo lnstitute of Technology, 
Jiaozuo City, Henan 454000 

T he People's Repu blic of China 
e-mail: guobaini@jzit.edu.cn 

A B STRACT 
In lhc pa¡>cr, t.hrcc prooís for an identity involving deriva t i ves o f a 

pOti.iti\'C dcfinit.c mntrix nncl its dct.erminant. a.re given using tcchniquc of 
hnca.r algebra. T hc idcnLity is bn.sic in diffcrential geomctry 

2000 Molhcmatia Sub1ect Clas:iification: ISA IS, I5A24. 
Keu worW ond phrwes: l clcntity, positi vo clefimtc malrix, d et.erminant , deri va t.i ve 

1 Int roductio n 

Let iW be an n-di me ns iona \1 n .$ 11 co nnected, C 1 Rie.mannia n ma n ifold . For 
d efin ition of manifold , piense re~ r t o s tandard texts fl, 4]. The Riema nnian 
metric on M associates t o each p E M an inner product o n Mp , which we 
d enote by ( , ). T he associat ed no rm will be deaoted by 1 I· T he Rieman nia n 
metn c LS C m 1.he s nse Lha l if X 1 Y a re C vector fields on M , l hen (X, Y} 
is n re.al-\'al ued funct iou on M . 

"Th~ au'bor WM s uppor tcd in pnrt by NNSF (# 10001016) of China., S F for Lhc Promincnt 
Youth ol Hcnan Provlnce, NS F of Hcnan Provincc (#~051800), F fo r P urc llcscnrdt of 
Nutur&I So0>Ct ol the Educnt lon DcpMtmcnt oí Hcnan Provioce (# 1999 11 000•1), Doctor 
F\md ol J..oauo lnft.ituLc of Tcchnology, China.. 

225 



226 Bai-Ni Guo 

Let U be an open set in M 1 and x ; U --t R" a diffeomorphism of U into 
IR" 1 lbat is , a cbart on M. Then associated to tbe chart are n coordinate 
vector 6elds , written as 8/8:.r) oras 8;, j = 1, ... 1 n . 

For Lhe given Ri emannian metric, define 

9;> = (8;,8>), 

g = detG, 

G = (g;>h9.> $•• 

a-' = (~'h9.> $• • 
where j , k = 1, . . . , n , det G and a- 1 denote tbe determinant and t he inverse 
oí G respectively. It is well-known that G is a positive definite mat rix . See {2 1 
pp. 3- 7J . 

The íollowing id entity involving de ri vatives of a posit ive defin ite matrix 
and its determfoant is fundamental in differentiaJ geometry. 

Theorem 1 Por 1 :S j ::; n, we have 

tr(G- 18;G) = 8;( 1og). (1) 

In th is s ho rl note , we wi ll g ive t hree proo fs of t he identit.y ( 1) us ing different 
technique oí linear a lge bra. Fa r concepta of linear a lgebra, picase refer to l3J. 

2 Three proofs of identity (1) 

Firs t proo í. Si.nce Lhe metr ic ma trix G = (9ij) is a pos itive definite matrix 1 
then we can assume its eigenvalues oí G are >.¡ > 0 1 i = 1, ... , n . From t heory 
o f linear a lge bra1 we ha.ve 

g = det G = IGI = I1 >. ;, (2) 
i= I 

lng = ¿in >.¡, (3) 
i = l 

" 8 ·>. · 
a;(lng) = I: T· 

i = l 1 
(4) 

where j :=:: 1, . .. , n . 
FU rlher, t here is a n orthogonal matrix P such t hat 

) = A, 

>-n 
(5) 



T hree proofs of an identity involviog ... 

therefore, ,.-e have G = PAP- 1 1 c- 1 = PA - 1 P - 1 1 and 
8, G = 8,(P/\P- 1 ) 

= (81P)/\P- 1 + P(8; A)P-1 + P/\(8,(P- 1)), 
c-1(8;G) = (P/\ - 1 p - ' )(8;P)/\P-1 + (P/\- 1 p - 1)(P(8,A)P- 1) 

+ (P/\ - 1 p - ')(P/\(8;(P- 1))) 
= pf1 - 1(P- 18¡P)/\P- 1 + P(/\ - 18, ¡ p - • + P8;(P- 1 ). 

l'rom p - • P = E, it follows t hat (8,(P - '))P + p - 1(8, P) = O, thus 

(6) 

(7) 

G 1(/J,G) = - (Pr1)[(8;( P- 1))PJ(PA - 1 ) - 1 + P( - 18, A)P- 1 + P8;(P- 1) 
= - (PA- 1 p - ')P[(8;(P- 1 ))PJP- 1(PA - 1 p -•¡-• 

+ P(/\ - 18; /\)P- ' + P8,(P- 1 ) 
= - G(P8,(P- 1 ))G- 1 + P(A- 18,A)P- ' + P8,(P- ' ). 

(8) 

sing 1he formula <r(AO) = t r(BA) , <r(P- 1 AP) = tr A, and tr(A + 8) = 
tr A+ tr B, we obtnin 

tr[G 1(8,G)J = tr(P81 (P 1)) + tr[P(1\ 18, A)P- 1J- tr!G(P8,(P- 1))G- 1J 

= tr(A - 18;1\) 

= t~ 
•= I A, 

= 8;(1ng) . 
(9) 

227 

Thc proof is complete. • 
R e ma.rk l In /oct, we have obtained th c /ollowing 

1r(G- 18;G) = tr[(8;G)G - 1J = 8,(ln !Gil= 8,(Jng). (10) 

t.."Co nd proof. We partilion lhe matri.x G by columns, lhat is 

G = (a, , .. . , crn ), (11 ) 

(12) 



228 B ai-Ni Guo 

where l .$; i .$; n . Tben we bave 

where 

8 · 1GI 
8;(1ng) =8;1nlGI = lcJ' 

n 

( 13) 

a, IGI = B; lo¡, ... ,o,,I = ¿ lcr1,. .. ·"•-1,8;0¡, 0¡+1 , ...• onl. (14) 
i = I 

(
8;9¡, ) 

8)0, = : 1 

819ni 

i = 1, 2, .. ,n. (15) 

The Laplace expansion y ields 

8; IGI = ¿ L (8;9ki)G;;, (16) 
i=l k= l 

where Gt. =e,,. is 1.he cofac t or of the e lemenL 9ik = 9k1 in symmetric maLrix 
e" = G. He nce 

.1 !l 
8;(1ng) = jGj .L (l!,g,,)G.,. 

1,k = l 

( 17) 

Moreove r, since 8iG = (8i9ik) a.nd c- 1 = fa¡ = ífml, where e· denotes 
the adjoint o f G, we have 

(G- '8 ·G) = t (G .. )(8;g;k) 
" ' r IGI 

( 18) 

1 .. 
= jGj L G.,(8;g., ), 

l ,k = I 

lhe 1den111¡· ir(G-18,G) = 8;(1 n IGI) follows. • 
R c mn.rk 2 For arbit,rary squarc matri::r A o/ arder n, i/ IAI > O, it.s element 
a,1 u o /uncl 1on o/ x, lhe r1 

d( ln IAI) = tr[A - ' dA] = tr( d;! A-1) . 
dx dx dx 

(19) 



T hree proofs of an identity invol ving ... 

R e m a.rk 3 Lct A(t ) ia an in ve rtible differentiable molru-, lh en 

(d el A)' = (d el A ) lr(A- 1 A'). 

where A' dt nott:J th e derivati ue o/ m atrix A unth rupo;t to t . 

229 

(20) 

T bird pro of. Let e· = (G11 ) d enote the adj oint of the positive d efi nit e 
mat rix G 1 lhen G,1 = G¡¡, nnd 

_, _ G' 8¡G _ t r(G' 81G) _ ~ ~ .. 
tr(G 8,G) - t r IGI - IGI - ¿_, c ,,(8,g,,), 

9 1.1- =- I 
(21} 

8 g 8 IGI l n l n 
8,{ln g ) = :E. = =.! = - 8; L Y11 G11 = - L: [{8,g,,)G 11 + 9118;G11]. 

9 g g l = l g l=I 
(22} 

The prooí red uces LO p rove t hnL 

(23) 
l -= 1 •-=2 1= 1 

lu íact 1 wc have 

¿g.,(8, G.,) = L L (8,g,¡)G,, , k = l , 2, ... , n. (2~ ) 
i.-=I 1j lc l= l 

T his compleLes t he prooí. • 
R efe r e n ces 

[IJ M P do Carmo1 Oifferential Geometry o/ Curves and S urfaces, Prc ntice-
llnll. loe., Englewood C liffs, Ncw Jersey, 1976. 

{2J 1 Cha\-el, E 19cnvalu es in Riema nruon Gcornctry, Acadcmic Press, 1984. 

[3J A Ramachandra H.ao and P. Bhi masankarnm, Lmeor Algebm, 2nd edition, 
TcxlS aod Readi ngs in Mathematics 19 , Hindustan Book Agency, New 
Delh1, lodi•, 2000. 

141 F W \\'aro r , Foundations o/ D1fferent1able Mantfolds and Lie Groups, 
Cradua~ Thxts in Mothemnt ics 9 4, pringer·Verlag, 1983. C hina Aca-
demte Pubhshcrs, Bcij ing, 19 3.