Page 211 - Determinants and Their Applications in Mathematical Physics

196 5. Further Determinant Theory

† rs † 2 2 rs
c r c s E = − (c + c )E
r s
r,s r,s
1
= − E rst,rst , (5.4.25)
3
r,s,t
1
2
c r c s E rs = c E rr − E rst,rst , (5.4.26)
3
r
r,s r r,s,t
1
2 2 2
(c + c )E rs =2 c E rr + E rst,rst , (5.4.27)
3
r s r
r,s r r,s,t
1
† 2
c E rs = E rst,rst + c r E rs,rs , (5.4.28)
6
r
r,s r,s,t r,s
1
† 2
c E rs = E rst,rst − c r E rs,rs . (5.4.29)
6
s
r,s r,s,t r,s
To prove (5.4.20), apply the second equation of (5.4.4) and (5.4.16).
E pr,ps = ∂E rs .
∂x p
Multiply by (c r − c s ) and sum over r and s:
∂
(c r − c s )E pr,ps = (c r − c s )E rs
r,s ∂x p r,s
∂
= E rs,rs
∂x p
r,s

= E prs,prs ,
r,s
which is equivalent to (5.4.20). The application of the fifth equation in
(5.4.4) with the modification (i, p, r, q, s) → (u, r, t, s, t) to (5.4.20) yields
(5.4.21).
To prove (5.4.22), sum (5.4.11) over i and j, change the dummy variables
as indicated
2 2
(c − c )E ij = F − G
i j
i,j
where, referring to (5.4.6) and (5.4.7),
       

F =  E is   c r E rj  +  E rj   c s E is 
i,s r,j r,j i,s
 
 
= E ii  c r E rj + c s E is 
 
i r,j i,s
(j→s) (i→r)