Page 61 - Determinants and Their Applications in Mathematical Physics

46 3. Intermediate Determinant Theory

The proof is completed by applying (C) and (A 1 ) with n → n−1. Theorem
3.7 is applied in Section 6.6 on the Matsukidaira–Satsuma equations.
Theorem 3.8.
(n+1) (n+1)
A H n = A H r ,
n+1,r n+1,n
where

(n−1)
A
(n+1)
A rr (n+1) (n)
rj − A A .
(n−1) (n+1) nj r,n−1;rn
H j =
A A
n−1,r n−1,j
Proof. Return to (3.6.18), multiply by A n+1 /A n and apply the Jacobi
identity:

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

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

(n) (n+1)
+ A A nr A nn =0,
(n+1)
A A (n+1)
r,n−1;rn
n+1,r n+1,n
(n+1) (n−1) (n+1) (n−1) (n+1) (n+1) (n)
A A A − A A − A A
n+1,r rr n−1,n n−1,r rn nn r,n−1;rn
(n+1) (n−1) (n+1) (n+1) (n−1) (n) (n+1)
= A A A − A A − A A ,
n+1,n rr n−1,r rr n−1,r r,n−1;rn nr
(n−1)

(n+1) (n+1) (n+1) (n)
A A rr A rn − A A
(n−1)
A A (n+1) nn r,n−1;rn
n+1,r
n−1,r n−1,n
(n−1)

(n+1) (n+1) (n+1) (n)
= A A rr A rr − A A .
(n−1)
A A
n+1,n (n+1) nr r,n−1;rn
n−1,r n−1,r
The theorem follows.
Exercise. Prove that
(n) (n) (n)
A A ... A A (n+1)
i 1 j 1 j 1 j 2 i 1 j r−1 i 1 ,n+1
(n) (n) (n) (n+1)
A A ... A A r−1 (n+1)

i 2 j 1 i 2 j 2 i 2 j r−1 i 2 ,n+1 = A A .
i 1 i 2 ...i r ;j 1 j 2 ...j r−1 ,n+1
n
...................................
(n) (n) (n) (n+1)
A A ... A A
i r j 1 i r j 2 i r j r−1 i r ,n+1 r
When r = 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
A n = |a ij | n

= C 1 C 2 C 3 ··· C n
n