Determinants and Their Applications in Mathematical Physics

44 3. Intermediate Determinant Theory

Proof. Denote the left side of variant (A) byE. Then, applying the

Jacobi identity,

An+1E=An+1

∣

∣

∣

∣

∣

A

(n)

ip

A

(n+1)

i,n+1

A

(n)

jp

A

(n+1)

j,n+1

∣

∣

∣

∣

∣

−An

∣

∣

∣

∣

∣

A

(n+1)

ip

A

(n+1)

i,n+1

A

(n+1)

jp

A

(n+1)

j,n+1

∣

∣

∣

∣

∣

=A

(n+1)

i,n+1

Fj−A

(n+1)

j,n+1

Fi, (3.6.13)

where

Fi=AnA

(n+1)

ip

−An+1A

(n)

ip

=

[∣

∣

∣

∣

∣

A

(n+1)

ip A

(n+1)

i,n+1

A

(n+1)

n+1,p

A

(n+1)

n+1,n+1

∣

∣

∣

∣

∣

−An+1A

(n)

ip

]

+A

(n+1)

i,n+1A

(n+1)

n+1,p

=A

(n+1)

i,n+1

A

(n+1)

n+1,p

.

Hence,

An+1E=

[

A

(n+1)

i,n+1

A

(n+1)

j,n+1

−A

(n+1)

j,n+1

A

(n+1)

i,n+1

]

A

(n+1)

n+1,p

=0. (3.6.14)

The result follows and variant (B) is proved in a similar manner. Variant

(A) appears in Section 4.8.5 on Turanians and is applied in Section 6.5.1

on Toda equations.

The proof of (C) applies a particular case of (A) and the Jacobi identity.

In (A), put (i, j, p)=(r, n, r):

∣

∣

∣

∣

∣

A

(n)

rr A

(n+1)

r,n+1

A

(n)

nr A

(n+1)

n,n+1

∣

∣

∣

∣

∣

−AnA

(n+1)

rn;r,n+1=0. (A^1 )

Denote the left side of (C) byP

AnP=An

∣

∣

∣

∣

A

(n)

rr A

(n+1)

rr

A

(n)

nr A

(n+1)

nr

∣

∣

∣

∣

−A

(n+1)

n+1,r

∣

∣

∣

∣

∣

A

(n)

rr A

(n+1)

r,n+1

A

(n)

nr A

(n+1)

n,n+1

∣ ∣ ∣ ∣ ∣ =

∣

∣

∣

∣

∣

∣

∣

A

(n)

rr A

(n+1)

rr A

(n+1)

r,n+1

A

(n)

nr A

(n+1)

nr A

(n+1)

n,n+1

• A

(n+1)

n+1,r

A

(n+1)

n+1,n+1

∣

∣

∣

∣

∣

∣

∣

=A

(n)

rr

Gn−A

(n)

nr

Gr, (3.6.15)

where

Gi=

∣

∣

∣

∣

∣

A

(n+1)

ir

A

(n+1)

i,n+1

A

(n+1)

n+1,r A

(n+1)

n+1,n+1

∣

∣

∣

∣

∣

=An+1A

(n+1)

i,n+1;r,n+1. (3.6.16)

Hence,

AnP=An+1

[

A

(n)

rr

A

(n+1)

n,n+1;r,n+1

−A

(n)

nr

A

(n+1)

r,n+1;r,n+1

]

.