Page 208 - Determinants and Their Applications in Mathematical Physics

5.4 The Cofactors of the Matsuno Determinant 193
∂

E + ∂x i E pru,qsv = e ipru,iqsv , (5.4.4)
ii
etc.

5.4.2 First Cofactors

When f r + g r = 0, the double-sum identities (C) and (D) in Section 3.4
become
n n

† rs †
(f r + g s )a rs A =0, (C )
r=1 s=1
n n

† is rj ij †
(f r + g s )a rs A A =(f i + g j )A . (D )
r=1 s=1
Applying (C )to E with f r = −g r = c ,
†
m
r
c − c m

n n
m

† r s rs
E =0. (5.4.5)
r=1 s=1 c r − c s
Putting m =1, 2, 3 yields the following particular cases:
! !
m =1: † E rs =0,
r s
which is equivalent to

E rs = E ; (5.4.6)
rr
r s r
! !
m =2: † (c r + c s )E rs =0,
r s
which is equivalent to

(c r + c s )E rs =2 c r E ; (5.4.7)
rr
r s r
2
2
m =3: ! ! † (c + c r c s + c )E rs =0,
r s
r s
which is equivalent to
2 2 2
(c + c r c s + c )E rs =3 c E . (5.4.8)
rr
r s r
r s r
†
Applying (D )to E, again with f r = −g r = c ,
m
r

m
c − c m
† r s is rj m m ij
E E =(c − c )E . (5.4.9)
i j
r s c r − c s
Putting m =1, 2 yields the following particular cases: