Determinants and Their Applications in Mathematical Physics

6.6 The Matsukidaira–Satsuma Equations 259

Hence applying the Jacobi identity (Section 3.6),

∣

∣

∣

∣

τr+2 τr+1

τr+1 τr

∣

∣

∣

∣

=

∣

∣

∣

∣

∣

A

(n+1)

11

(r)(−1)

n

A

(n+1)

1 ,n+1

(r)

(−1)

n

A

(n+1)

n+1, 1 (r) A

(n+1)

n+1,n+1(r)

∣

∣

∣

∣

∣

=A

(n+1)

(r)A

(n−1)

(r+2).

Replacingrbyr−1,

∣

∣

∣

∣

τr+1 τr

τr τr− 1

∣

∣

∣

∣

=A

(n+1)

(r−1)A

(n−1)

(r+ 1) (6.6.3)

τ

′

r=−A

(n+1)

n,n+1(r)

=−A

(n+1)

n+1,n

(r)

τ

′′

r=A

(n+1)

nn (r).

Hence,

∣

∣

∣

∣

τ

′′

r τ

′

r

τ

′

r τr

∣

∣

∣

∣

=

∣

∣

∣

∣

∣

A

(n+1)

nn (r) A

(n+1)

n,n+1

(r)

A

(n+1)

n+1,n

(r) A

(n+1)

n+1,n+1

(r)

∣

∣

∣

∣

∣

=A

(n+1)

(r)A

(n+1)

n,n+1;n,n+1(r)

=A

(n+1)

(r)A

(n−1)

(r). (6.6.4)

Similarly,

τr+1=−A

(n+1)

1 ,n+1

(r)

=−A

(n+1)

n+1, 1 (r),

τ

′

r+1

=(−1)

n+1

A

(n+1)

1 n

(r),

∣

∣

∣

∣

τ

′

r+1

τr+1

τ

′

r

τr

∣

∣

∣

∣

=A

(n+1)

(r)A

(n−1)

(r+1). (6.6.5)

Replacingrbyr−1,

∣

∣

∣

∣

τ

′

r τr

τ

′

r− 1

τr− 1

∣

∣

∣

∣

=A

(n+1)

(r−1)A

(n−1)

(r). (6.6.6)

Theorem 6.7 follows from (6.6.3)–(6.6.6).

Theorem 6.8.

∣ ∣ ∣ ∣ ∣ ∣

τr τr+1 τ

′

r+1

τr− 1 τr τ

′

r

τ

′

r− 1

τ

′

r

τ

′′

r

∣ ∣ ∣ ∣ ∣ ∣

=0.

Proof. Denote the determinant by F. Then, Theorem 6.7 can be

expressed in the form

F 33 F 11 =F 31 F 13.