46 3. Intermediate Determinant Theory
The proof is completed by applying (C) and (A 1 ) withn→n−1. Theorem
3.7 is applied in Section 6.6 on the Matsukidaira–Satsuma equations.
Theorem 3.8.
A
(n+1)
n+1,r
Hn=A
(n+1)
n+1,n
Hr,
where
Hj=
∣
∣
∣
∣
∣
A
(n−1)
rr A
(n+1)
rj
A
(n−1)
n− 1 ,r
A
(n+1)
n− 1 ,j
∣
∣
∣
∣
∣
−A
(n+1)
nj A
(n)
r,n−1;rn.
Proof. Return to (3.6.18), multiply byAn+1/Anand apply the Jacobi
identity:
A
(n−1)
n− 1 ,r
∣
∣
∣
∣
A
(n+1)
rr A
(n+1)
rn
A
(n+1)
n+1,r
A
(n+1)
n+1,n
∣
∣
∣
∣
−A
(n−1)
rr
∣
∣
∣
∣
∣
A
(n+1)
n− 1 ,r
A
(n+1)
n− 1 ,n
A
(n+1)
n+1,r A
(n+1)
n+1,n
∣
∣
∣
∣
∣
+A
(n)
r,n−1;rn
∣
∣
∣
∣
A
(n+1)
nr A
(n+1)
nn
A
(n+1)
n+1,r
A
(n+1)
n+1,n
∣
∣
∣
∣
=0,
A
(n+1)
n+1,r
[
A
(n−1)
rr A
(n+1)
n− 1 ,n
−A
(n−1)
n− 1 ,r
A
(n+1)
rn −A
(n+1)
nn A
(n)
r,n−1;rn
]
= A
(n+1)
n+1,n
[
A
(n−1)
rr
A
(n+1)
n− 1 ,r
−A
(n+1)
rr
A
(n−1)
n− 1 ,r
−A
(n)
r,n−1;rn
A
(n+1)
nr
]
,
A
(n+1)
n+1,r
[∣
∣
∣
∣
A
(n−1)
rr A
(n+1)
rn
A
(n−1)
n− 1 ,r A
(n+1)
n− 1 ,n
∣
∣
∣
∣
−A
(n+1)
nn A
(n)
r,n−1;rn
]
= A
(n+1)
n+1,n
[∣
∣
∣
∣
A
(n−1)
rr A
(n+1)
rr
A
(n−1)
n− 1 ,r A
(n+1)
n− 1 ,r
∣
∣
∣
∣
−A
(n+1)
nr A
(n)
r,n−1;rn
]
The theorem follows.
Exercise.Prove that
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
A
(n)
i 1 j 1
A
(n)
j 1 j 2
... A
(n)
i 1 jr− 1
A
(n+1)
i 1 ,n+1
A
(n)
i 2 j 1
A
(n)
i 2 j 2
... A
(n)
i 2 jr− 1
A
(n+1)
i 2 ,n+1
A
(n)
irj 1
A
(n)
irj 2
... A
(n)
irjr− 1
A
(n+1)
ir,n+1
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ r
=A
r− 1
n A
(n+1)
i 1 i 2 ...ir;j 1 j 2 ...jr− 1 ,n+1.
Whenr= 2, this identity degenerates into Variant (A). Generalize Variant
(B) in a similar manner.
3.7 Bordered Determinants
3.7.1 Basic Formulas; The Cauchy Expansion
Let
An=|aij|n
=
∣
∣C
1 C 2 C 3 ···Cn
∣
∣
n