Determinants and Their Applications in Mathematical Physics

5.4 The Cofactors of the Matsuno Determinant 197

=

∑

i

E

ii

∑

r,s

(cr+cs)E

rs

=2

∑

i

E

ii

∑

r

crE

rr

, (5.4.30)

G=2

∑

i,j,r

crE

ir

E

rj

. (5.4.31)

Modify (5.4.10) withj=iby making the changesi↔rands→j. This

gives

G=2

∑

r

cr

∑

i

E

ir

E

ri

. (5.4.32)

Hence,

∑

i,j

(c

2

i−c

2

j)E

ij

=2

∑

i,r

cr

∣

∣

∣

∣

E

ii

E

ir

E

ri

E

rr

∣

∣

∣

∣

=2

∑

i,r

E

ir,ir

,

which is equivalent to (5.4.22).

To prove (5.4.23) multiply (5.4.10) by (ci−cj), sum overiandj, change

the dummy variables as indicated, and refer to (5.4.6):

∑

i,j

(ci−cj)

2

E

ij

=H−J, (5.4.33)

where

H=

∑

i,j

(ci−cj)

∑

r,s

E

is

E

rj

=





∑

r,j

E

rj













∑

i,s

(s→j)

ciE

is









−





∑

i,s

E

is













∑

r,j

(r→i)

cjE

rj









=

∑

r

E

rr

∑

i,j

(ci−cj)E

ij

, (5.4.34)

J=

∑

i,j

(ci−cj)

∑

r

E

ir

E

rj

. (5.4.35)

Hence, referring to (5.4.20) with suitable changes in the dummy variables,

∑

i,j

(ci−cj)

2

E

ij

=

∑

i,j,r

(ci−cj)

∣

∣

∣

∣

E

ij

E

ir

E

rj

E

rr

∣

∣

∣

∣

=

∑

i,j,r

(ci−cj)E

ir,jr