Determinants and Their Applications in Mathematical Physics

200 5. Further Determinant Theory

=U

2

−V

2

+

∑

r

∂S

∂xr

, (5.4.42)

where

S=

∑

i,j

cicjE

ij

. (5.4.43)

This function is identical to the left-hand side of (5.4.26). Let

T=

∑

i,j,r,s

(ci+cj)(cr+cs)E

is

E

rj

. (5.4.44)

Then, applying (5.4.6),

T=

∑

i,s

ciE

is

∑

j,r

crE

rj

+

∑

j,r

E

rj

∑

i,s

cicsE

is

+

∑

i,s

E

is

∑

j,r

cjcrE

rj

+

∑

j,r

cjE

rj

∑

i,s

csE

is

=(U+V)

2

+2S

∑

r,s

E

rs

+(U−V)

2

=2(U

2

+V

2

)+2S

∑

r

E

rr

. (5.4.45)

EliminatingVfrom (5.4.42),

T+2R=4U

2

+2

∑

r

(

E

rr

+

∂

∂xr

)

S. (5.4.46)

To obtain a formula forQ, multiply (5.4.11) by (ci+cj), sum overiand

j, and apply (5.4.13) with the modifications (i, j)↔(r, s) on the left and

(i, r)→(r, s) on the right:

Q=

∑

i,j,r,s

(ci+cj)(cr+cs)E

is

E

rj

− 2

∑

i,j,r

cr(ci+cj)E

ir

E

rj

=T− 2

∑

r

cr

∑

i,j

(ci+cj)E

ir

E

rj

=T− 4

∑

r,s

crcsE

rs

E

sr

. (5.4.47)

EliminatingTfrom (5.4.46) and applying (5.4.26) and the fourth and sixth

lines of (5.4.4),

Q+2R=4

∑

r,s

crcs(E

rr

E

ss

−E

sr

E

rs

)

+2

∑

r

(

E

rr

+

∂

∂xr

)

[

∑

s

c

2

s

E

ss

−

1

3

∑

s,t,u

E

stu,stu

]