Determinants and Their Applications in Mathematical Physics

198 5. Further Determinant Theory

=

∑

i,j,r

E

ijr,ijr

,

which is equivalent to (5.4.23). The application of a suitably modified the

fourth line of (5.4.4) to (5.4.23) yields (5.4.24). Identities (5.4.27)–(5.4.29)

follow from (5.4.8), (5.4.22), (5.4.24), and the identities

3 crcs=(c

2

r

+crcs+c

2

s

)−(cr−cs)

2

,

6 c

2

r

=2(c

2

r

+crcs+c

2

s

)+(cr−cs)

2

+3(c

2

r

−c

2

s

),

6 c

2

s

=2(c

2

r

+crcs+c

2

s

)+(cr−cs)

2

−3(c

2

r

−c

2

s

).

5.4.5 Three Further Identities

The identities

∑

r,s

(c

2

r+c

2

s)(cr−cs)E

rs

=2

∑

r,s

c

2

rE

rs,rs

+

1

3

∑

r,s,u,v

E

rsuv,rsuv

, (5.4.36)

∑

r,s

(c

2

r

−c

2

s

)(cr+cs)E

rs

=2

∑

r,s

cr(cr+cs)E

rs,rs

−

1

6

∑

r,s,u,v

E

rsuv,rsuv

, (5.4.37)

∑

r,s

crcs(cr−cs)E

rs

=

∑

r,s

crcsE

rs,rs

−

1

4

∑

r,s,u,v

E

rsuv,rsuv

(5.4.38)

are more difficult to prove than those in earlier sections. The last one has

an application in Section 6.8 on the KP equation, but its proof is linked to

those of the other two.

Denote the left sides of the three identities byP,Q, andR, respectively.

To prove (5.4.36), multiply the second equation in (5.4.10) by (c

2

i+c

2

j),

sum overiandjand refer to (5.4.4), (5.4.6), and (5.4.27):

P=

∑

i,j,r,s

(c

2

i

+c

2

j

)E

is

E

rj

+

∑

i,j,r

(c

2

i

+c

2

j

)E

ir

E

rj

=





∑

j,r

E

rj













∑

i,s

(s→j)

c

2

iE

is









+





∑

i,s

E

is













∑

j,r

(r→i)

c

2

jE

rj









+

∑

i,j,r

(c

2

i+c

2

j)

∂E

ij

∂xr