Determinants and Their Applications in Mathematical Physics

194 5. Further Determinant Theory

m=1:

∑

r

∑

s

†

E

is

E

rj

=(ci−cj)E

ij

,

which is equivalent to

∑

r

∑

s

E

is

E

rj

−

∑

r

E

ir

E

rj

=(ci−cj)E

ij

; (5.4.10)

m=2:

∑

r

∑

s

†

(cr+cs)E

is

E

rj

=(c

2

i−c

2

j)E

ij

,

which is equivalent to

∑

r

∑

s

(cr+cs)E

is

E

rj

− 2

∑

r

crE

ir

E

rj

=(c

2

i−c

2

j)E

ij

, (5.4.11)

etc. Note that the right-hand side of (5.4.9) is zero whenj=ifor all values

ofm. In particular, (5.4.10) becomes

∑

r

∑

s

E

is

E

ri

=

∑

r

E

ir

E

ri

(5.4.12)

and the equation in itemm= 2 becomes

∑

r

∑

s

(cr+cs)E

is

E

ri

=2

∑

r

crE

ir

E

ri

. (5.4.13)

5.4.3 First and Second Cofactors..............

The following five identities relate the first and second cofactors ofE: They

all remain valid when the parameters are lowered.

∑

r,s

†

E

ir,js

=−(ci−cj)E

ij

, (5.4.14)

∑

r,s

†

(cr+cs)E

ir,js

=−(c

2

i−c

2

j)E

ij

, (5.4.15)

∑

r,s

(cr−cs)E

rs

=

∑

r,s

E

rs,rs

, (5.4.16)

2

∑

r,s

†

crE

rs

=− 2

∑

r,s

†

csE

rs

=

∑

r,s

E

rs,rs

, (5.4.17)

∑

r<s

(csE

rs

+crE

sr

+E

rs,rs

)=0. (5.4.18)

To prove (5.4.14), apply the Jacobi identity toE

ir,js

and refer to (5.4.6)

and the equation in itemm=1.

∑

r,s

†

E

ir,js

=

∑

r,s

†

∣

∣

∣

∣

E

ij

E

is

E

rj

E

rs

∣

∣

∣

∣

=E

ij

∑

r,s

†

E

rs

−

∑

r,s

†

E

is

E

rj