Page 59 - Determinants and Their Applications in Mathematical Physics

44 3. Intermediate Determinant Theory

Proof. Denote the left side of variant (A) by E. Then, applying the
Jacobi identity,

(n) (n+1)
A A (n+1) A A (n+1)

ip i,n+1 ip i,n+1
(n) (n+1) − A n (n+1) (n+1)
A n+1 E = A n+1
A A A A
j,n+1 j,n+1
jp jp
(n+1) (n+1)
= A F j − A F i , (3.6.13)
i,n+1 j,n+1
where
(n+1) (n)
F i = A n A − A n+1 A
ip ip

(n+1)
A A (n+1) (n) (n+1) (n+1)

ip i,n+1 − A n+1 A + A A
= (n+1) (n+1) ip i,n+1 n+1,p
A A n+1,n+1
n+1,p
(n+1) (n+1)
= A i,n+1 A n+1,p .
Hence,
(n+1) (n+1) (n+1) (n+1) (n+1)
A n+1 E = A A − A A A
i,n+1 j,n+1 j,n+1 i,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):

(n)
A
(n+1)
r,n+1 − A n A =0. (A 1 )
A rr (n+1)
(n) (n+1)
rn;r,n+1
A nr A n,n+1
Denote the left side of (C) by P

(n) (n+1)
(n)
(n+1) (n+1) A rr A
− A r,n+1
A rr
A rr
(n) (n) (n+1)

A
n+1,r
A n P = A n
(n+1)
A nr n,n+1
A nr
A nr
(n) (n+1) (n+1)

A
A rr A rr r,n+1
(n) (n+1)
A
(n+1)
n,n+1
= A nr
A nr
(n+1) (n+1)
• A A n+1,n+1

n+1,r
= A (n) G n − A (n) G r , (3.6.15)
rr nr
where
(n+1)
A A (n+1)

ir i,n+1
(n+1) (n+1)
G i =
A A
n+1,n+1
n+1,r
(n+1)
= A n+1 A . (3.6.16)
i,n+1;r,n+1
Hence,
(n) (n+1) (n) (n+1)
A n P = A n+1 A A n,n+1;r,n+1 − A A r,n+1;r,n+1 .
rr nr