Page 274 - Determinants and Their Applications in Mathematical Physics

6.6 The Matsukidaira–Satsuma Equations 259

Hence applying the Jacobi identity (Section 3.6),

(n+1) (n+1)
τ A (r) (−1) A
n
r+2 τ r+1 11 1,n+1 (r)

(n+1) (n+1)
=
τ r+1 (−1) A (r) A (r)

n
n+1,1 n+1,n+1
τ r
= A (n+1) (r)A (n−1) (r +2).
Replacing r by r − 1,

τ
r+1 τ r = A (n+1) (r − 1)A (n−1) (r + 1) (6.6.3)

τ r−1

τ r
(n+1)
τ = −A (r)

r n,n+1
(n+1)
= −A (r)
n+1,n

τ = A (n+1) (r).
r nn
Hence,

(n+1) (n+1)
τ τ (r) A
n,n+1
A nn (r)
r r
τ = (n+1) (n+1)

A (r) A (r)
r τ r n+1,n+1
n+1,n
= A (n+1) (r)A (n+1) (r)
n,n+1;n,n+1
= A (n+1) (r)A (n−1) (r). (6.6.4)
Similarly,
(n+1)
τ r+1 = −A (r)
1,n+1
(n+1)
= −A (r),
n+1,1

τ r+1 =(−1) n+1 A (n+1) (r),
1n
τ
(n+1) (n−1)
r+1 τ r+1 = A (r)A (r +1). (6.6.5)
τ

r τ r
Replacing r by r − 1,

τ (n+1) (n−1)
r τ r
= A (r − 1)A (r). (6.6.6)

τ r−1 τ r−1
Theorem 6.7 follows from (6.6.3)–(6.6.6).
Theorem 6.8.

τ r+1
τ
τ r r+1
τ
r−1 τ r τ r =0.

τ τ τ
r−1 r r

Proof. Denote the determinant by F. Then, Theorem 6.7 can be
expressed in the form
F 33 F 11 = F 31 F 13 .