Page 209 - Determinants and Their Applications in Mathematical Physics

194 5. Further Determinant Theory

m =1: ! ! † E E rj =(c i − c j )E ,
is
ij
r s
which is equivalent to

E E rj − E E rj =(c i − c j )E ; (5.4.10)
ij
is
ir
r s r
2
2
m =2: ! ! † (c r + c s )E E rj =(c − c )E ,
is
ij
i j
r s
which is equivalent to
2 2
(c r + c s )E E rj − 2 c r E E rj =(c − c )E , (5.4.11)
ij
ir
is
i j
r s r
etc. Note that the right-hand side of (5.4.9) is zero when j = i for all values
of m. In particular, (5.4.10) becomes

E E ri = E E ri (5.4.12)
ir
is
r s r
and the equation in item m = 2 becomes

(c r + c s )E E ri =2 c r E E . (5.4.13)
is
ir
ri
r s r
5.4.3 First and Second Cofactors
The following five identities relate the first and second cofactors of E: They
all remain valid when the parameters are lowered.

† ir,js ij
E = −(c i − c j )E , (5.4.14)
r,s
2 2
† ir,js ij
(c r + c s )E = −(c − c )E , (5.4.15)
i j
r,s

(c r − c s )E rs = E rs,rs , (5.4.16)
r,s r,s

2 † c r E rs = −2 † c s E rs = E rs,rs , (5.4.17)
r,s r,s r,s

(c s E rs + c r E sr + E rs,rs )=0. (5.4.18)
r<s
To prove (5.4.14), apply the Jacobi identity to E ir,js and refer to (5.4.6)
and the equation in item m =1.

E ij E is
† ir,js
E = †
E rj E
rs
r,s r,s

= E ij † E rs − † E E rj
is
r,s r,s