Determinants and Their Applications in Mathematical Physics

5.4 The Cofactors of the Matsuno Determinant 193

(

E

ii

+

∂

∂xi

)

E

pru,qsv

=e

ipru,iqsv

, (5.4.4)

etc.

5.4.2 First Cofactors.....................

Whenfr+gr= 0, the double-sum identities (C) and (D) in Section 3.4

become

n

∑

r=1

n

∑

s=1

†

(fr+gs)arsA

rs

=0, (C

†

)

n

∑

r=1

n

∑

s=1

†

(fr+gs)arsA

is

A

rj

=(fi+gj)A

ij

. (D

†

)

Applying (C
†
)toEwithfr=−gr=c
m
r

,

n

∑

r=1

n

∑

s=1

†

(

c

m

r

−c

m

s

cr−cs

)

E

rs

=0. (5.4.5)

Puttingm=1, 2 ,3 yields the following particular cases:

m=1:

∑

r

∑

s

†

E

rs

=0,

which is equivalent to

∑

r

∑

s

E

rs

=

∑

r

E

rr

; (5.4.6)

m=2:

∑

r

∑

s

†

(cr+cs)E

rs

=0,

which is equivalent to

∑

r

∑

s

(cr+cs)E

rs

=2

∑

r

crE

rr

; (5.4.7)

m=3:

∑

r

∑

s

†

(c

2

r+crcs+c

2

s)E

rs

=0,

which is equivalent to

∑

r

∑

s

(c

2

r

+crcs+c

2

s

)E

rs

=3

∑

r

c

2

r

E

rr

. (5.4.8)

Applying (D

†

)toE, again withfr=−gr=c

m

r,

∑

r

∑

s

†

(

c

m

r −c

m

s

cr−cs

)

E

is

E

rj

=(c

m

i

−c

m

j

)E

ij

. (5.4.9)

Puttingm=1,2 yields the following particular cases: