Determinants and Their Applications in Mathematical Physics

250 6. Applications of Determinants in Mathematical Physics

(A

ij

)

′

=−

∑

r

e

cru

A

rj

∑

s

e

csu

A

is

, (6.4.4)

2

∑

r

brcrA

rr

+

∑

r,s

e

(cr+cs)u

A

rs

=2

∑

r

cr, (6.4.5)

2

∑

r

brcrA

ir

A

rj

+

∑

r

e

cru

A

rj

∑

s

e

csu

A

is

=(ci+cj)A

ij

. (6.4.6)

Put

φi=

∑

s

e

csu

A

is

. (6.4.7)

Then (6.4.4) and (6.4.6) become

(A

ij

)

′

=−φiφj, (6.4.8)

2

∑

r

brcrA

ir

A

rj

+φiφj=(ci+cj)A

ij

. (6.4.9)

Eliminating theφiφjterms,

(A

ij

)

′

+(ci+cj)A

ij

=2

∑

r

brcrA

ir

A

rj

,

[

e

(ci+cj)u

A

ij

]′

=2e

(ci+cj)u

∑

r

brcrA

ir

A

rj

. (6.4.10)

Differentiating (6.4.3),

(logA)

′′

=

∑

i,j

[

e

(ci+cj)u

A

ij

]′

=2

∑

r

brcr

∑

i

e

ciu

A

ir

∑

j

e

cju

A

rj

=2

∑

r

brcrφ

2

r

. (6.4.11)

Replacingsbyrin (6.4.7),

e

ciu

φi=

∑

r

e

(ci+cr)u

A

ir

,

(

e

cju

φi

)′

=2

∑

r

brcr

(

e

ciu

A

ir

)∑

j

e

cju

A

rj

=2

∑

r

brcrφre

ciu

A

ir

,

φ

′

i+ciφi=2

∑

r

brcrφrA

ir

.

Interchangeiandr, multiply bybrcrA
rj
, sum overr, and refer to (6.4.9):

∑

r

brcrA

rj

(φ

′

r

+crφr)=2

∑

i

biciφi

∑

r

brcrA

ir

A

rj