Page 213 - Determinants and Their Applications in Mathematical Physics

198 5. Further Determinant Theory

= E ijr,ijr ,
i,j,r
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
2
2
2
3c r c s =(c + c r c s + c ) − (c r − c s ) ,
r s
2
2
2
2
2
2
6c =2(c + c r c s + c )+(c r − c s ) +3(c − c ),
r r s r s
2
2
2
2
2
2
6c =2(c + c r c s + c )+(c r − c s ) − 3(c − c ).
s r s r s
5.4.5 Three Further Identities
The identities
2 2 2
(c + c )(c r − c s )E rs =2 c E rs,rs
r s r
r,s r,s
1
+ E rsuv,rsuv , (5.4.36)
3
r,s,u,v
2 2
(c − c )(c r + c s )E rs =2 c r (c r + c s )E rs,rs
r s
r,s r,s
1
− E rsuv,rsuv , (5.4.37)
6
r,s,u,v

c r c s (c r − c s )E rs = c r c s E rs,rs
r,s r,s
1
− E rsuv,rsuv (5.4.38)
4
r,s,u,v
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 by P, Q, and R, respectively.
2
2
To prove (5.4.36), multiply the second equation in (5.4.10) by (c + c ),
i j
sum over i and j and refer to (5.4.4), (5.4.6), and (5.4.27):
2 2 2 2
P = (c + c )E E rj + (c + c )E E rj
ir
is
i j i j
i,j,r,s i,j,r
   
   
 2   2 
=  E rj  c E is  +  E is  c E rj 


i j
   
j,r i,s i,s j,r
(s→j) (r→i)
2 2 ∂E ij
+ (c + c )
i j
i,j,r ∂x r