Page 212 - Determinants and Their Applications in Mathematical Physics

5.4 The Cofactors of the Matsuno Determinant 197

= E ii (c r + c s )E rs
r,s
i

=2 E ii c r E , (5.4.30)
rr
i r

G =2 c r E E . (5.4.31)
ir
rj
i,j,r
Modify (5.4.10) with j = i by making the changes i ↔ r and s → j. This
gives

G =2 c r E E . (5.4.32)
ir
ri
r
i
Hence,

2 2 E ii E ir
(c − c )E ij =2
E E rr
i j c r ri
i,j i,r

=2 E ir,ir ,
i,r
which is equivalent to (5.4.22).
To prove (5.4.23) multiply (5.4.10) by (c i −c j ), sum over i and j, change
the dummy variables as indicated, and refer to (5.4.6):
2
(c i − c j ) E ij = H − J, (5.4.33)
i,j
where

H = (c i − c j ) E E rj
is
i,j r,s
   
   
   
=  E rj  c i E is  −  E is  c j E rj 


   
r,j i,s i,s r,j
(s→j) (r→i)

= E rr (c i − c j )E , (5.4.34)
ij
r i,j

J = (c i − c j ) E E . (5.4.35)
ir
rj
r
i,j
Hence, referring to (5.4.20) with suitable changes in the dummy variables,

2 E ij E ir
(c i − c j ) E ij = (c i − c j )
E rj E rr
i,j i,j,r

= (c i − c j )E ir,jr
i,j,r