Determinants and Their Applications in Mathematical Physics

5.4 The Cofactors of the Matsuno Determinant 199

=

∑

r

E

rr

∑

i,j

(c

2

i+c

2

j)E

ij

+

∑

r

∂

∂xr

∑

i,j

(c

2

i+c

2

j)E

ij

=

∑

r

(

E

rr

+

∂

∂xr

)

∑

i,j

(c

2

i

+c

2

j

)E

ij

=

∑

v

(

E

vv

+

∂

∂xv

)

[

2

∑

r

c

2

rE

rr

+

1

3

∑

r,s,t

E

rst,rst

]

=2

∑

r,v

c

2

rE

rv,rv

+

1

3

∑

r,s,t,v

E

rstv,rstv

,

which is equivalent to (5.4.36).

Since

(c

2

r

−c

2

s

)(cr+cs)− 2 crcs(cr−cs)=(c

2

r

+c

2

s

)(cr−cs),

it follows immedately that

Q− 2 R=P. (5.4.39)

A second relation betweenQandRis found as follows. Let

U=

∑

r

crE

rr

,

V=

1

2

∑

r,s

E

rs,rs

. (5.4.40)

It follows from (5.4.17) that

V=

∑

r,s

crE

rs

−

∑

r

crE

rr

=

∑

r

crE

rr

−

∑

r,s

csE

rs

.

Hence

∑

r,s

crE

rs

=U+V,

∑

r,s

csE

rs

=U−V. (5.4.41)

To obtain a formula forR, multiply (5.4.10) bycicj, sum overiandj, and

apply the third equation of (5.4.4):

R=

∑

i,j,r,s

cicjE

is

E

rj

−

∑

i,j,r

cicjE

ir

E

rj

=





∑

i,s

ciE

is









∑

j,r

cjE

rj



+

∑

r

∂

∂xr

∑

i,j

cicjE

ij