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)
rjA(n−1)
n− 1 ,rA
(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)
rnA(n+1)
n+1,rA
(n+1)
n+1,n∣
∣
∣
∣
−A
(n−1)
rr∣
∣
∣
∣
∣
A
(n+1)
n− 1 ,rA
(n+1)
n− 1 ,nA
(n+1)
n+1,r A(n+1)
n+1,n∣
∣
∣
∣
∣
+A
(n)
r,n−1;rn∣
∣
∣
∣
A
(n+1)
nr A(n+1)
nnA(n+1)
n+1,rA
(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 ,rA
(n+1)
rn −A(n+1)
nn A(n)
r,n−1;rn]
= A
(n+1)
n+1,n[
A
(n−1)
rrA
(n+1)
n− 1 ,r−A
(n+1)
rrA
(n−1)
n− 1 ,r−A
(n)
r,n−1;rnA
(n+1)
nr]
,
A
(n+1)
n+1,r[∣
∣
∣
∣
A
(n−1)
rr A(n+1)
rnA(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)
rrA(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 1A
(n)
j 1 j 2... A
(n)
i 1 jr− 1A
(n+1)
i 1 ,n+1A
(n)
i 2 j 1A
(n)
i 2 j 2... A
(n)
i 2 jr− 1A
(n+1)
i 2 ,n+1A
(n)
irj 1A
(n)
irj 2... A
(n)
irjr− 1A
(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