Determinants and Their Applications in Mathematical Physics

278 6. Applications of Determinants in Mathematical Physics

Applying (A),

v=Dx(logA)=−

∑

r

λrerA

rr

, (6.8.3)

Dy(logA)=

∑

r

λrμrerA

rr

, (6.8.4)

Dt(logA)=4

∑

r

λr(b

2

r−brcr+c

2

r)erA

rr

. (6.8.5)

Applying (B),

Dx(A

ij

)=

∑

r

λrerA

ir

A

rj

, (6.8.6)

Dy(A

ij

)=−

∑

r

λrμrerA

ir

A

rj

, (6.8.7)

Dt(A

ij

)=− 4

∑

r

λr(b

2

r−brcr+c

2

r)erA

ir

A

rj

. (6.8.8)

Applying (C) with

i.fr=br,gr=cr;

ii.fr=b

2

r,gr=−c

2

r;

iii.fr=b

3

r,gr=c

3

r;

in turn,

∑

r

λrerA

rr

+

∑

r,s

A

rs

=

∑

r

λr, (6.8.9)

∑

r

λrμrerA

rr

+

∑

r,s

(br−cs)A

rs

=

∑

r

λrμr, (6.8.10)

∑

r

λr(b

2

r−brcr+c

2

r)erA

rr

+

∑

r,s

(b

2

r−brcs+c

2

s)A

rs

=

∑

r

λr(b

2

r

−brcr+c

2

r

).(6.8.11)

Applying (D) with (i)–(iii) in turn,

∑

r

λrerA

ir

A

rj

+

∑

r,s

A

is

A

rj

=(bi+cj)A

ij

, (6.8.12)

∑

r

λrμrerA

ir

A

rj

+

∑

r,s

(br−cs)A

is

A

rj

=(b

2

i

−c

2

j

)A

ij

, (6.8.13)

∑

r

λr(b

2

r−brcr+c

2

r)erA

ir

A

rj

+

∑

r,s

(b

2

r−brcs+c

2

s)A

is

A

rj

=(b

3

i

+c

3

j

)A

ij

. (6.8.14)

Eliminating the sum common to (6.8.3) and (6.8.9), the sum common to

(6.8.4) and (6.8.10) and the sum common to (6.8.5) and (6.8.11), we find