Page 210 - Determinants and Their Applications in Mathematical Physics

5.4 The Cofactors of the Matsuno Determinant 195

= −(c i − c j )E .
ij
Equation (5.4.15) can be proved in a similar manner by appling (5.4.7) and
the equation in item m = 2. The proof of (5.4.16) is a little more difficult.
Modify (5.4.12) by making the following changes in the parameters. First
i → k, then (r, s) → (i, j), and, finally, k → r. The result is

E E = E E . (5.4.19)
† rj ir ri ir
i,j i
Now sum (5.4.10) over i, j and refer to (5.4.19) and (5.4.6):
 

(c i − c j )E ij = E E rj −  E E rj 
is
ir
i,j i,j,r,s r i,j
   

=  E is   E rj  − E E ir
ri
i,s r,j r i

= E ii E rr − E E ir
ri
i r i,r

E ii E ir

=
E ri E rr
i,r

= E ir,ir ,
i,r
which is equivalent to (5.4.16). The symbol † can be attached to the sum
on the left without affecting its value. Hence, this identity together with
(5.4.7) yields (5.4.17), which can then be expressed in the symmetric form
(5.4.18) in which r< s.

5.4.4 Third and Fourth Cofactors

The following identities contain third and fourth cofactors of E:

(c r − c s )E rt,st = E rst,rst , (5.4.20)
r,s r,s

(c r − c s )E rtu,stu = E rstu,rstu , (5.4.21)
r,s r,s
2 2
(c − c )E rs =2 c r E rs,rs , (5.4.22)
r s
r,s r,s
2
(c r − c s ) E rs = E rst,rst , (5.4.23)
r,s r,s,t
2
(c r − c s ) E ru,su = E rstu,rstu , (5.4.24)
r,s r,s,t