Page 53 - Determinants and Their Applications in Mathematical Physics

38 3. Intermediate Determinant Theory

Hence, removing the factor A n from each row,

δ
|c ij | n = A n ij x i + H ij
n

A n n
which yields the stated result.
This theorem is applied in Section 6.7.4 on the K dV equation.
3.6 The Jacobi Identity and Variants

3.6.1 The Jacobi Identity — 1

Given an arbitrary determinant A = |a ij | n , the rejecter minor M p 1 p 2 ...p r ;q 1 q 2 ...q r
of order r are
of order (n − r) and the retainer minor N p 1 p 2 ...p r ;q 1 q 2 ...q r
defined in Section 3.2.1.
Define the retainer minor J of order r as follows:

J = J p 1 p 2 ...p r ;q 1 q 2 ...q r = adj N p 1 p 2 ...p r ;q 1 q 2 ...q r

···
A p 1 q 1 A p 2 q 1 A p r q 1

···
A p 1 q 2 A p 2 q 2 A p r q 2
. (3.6.1)
=
.........................

A p 1 q r A p 2 q r ··· A p r q r r
J is a minor of adj A. For example,
J 23,24 = adj N 23,24

a 22
= adj a 24
a 32 a 34

A 22
= A 32 .
A 24 A 34

The Jacobi identity on the minors of adj A is given by the following theorem:
Theorem.
= A r−1 , 1 ≤ r ≤ n − 1.
J p 1 p 2 ...p r ;q 1 q 2 ...q r M p 1 p 2 ...p r ;q 1 q 2 ...q r
Referring to the section on the cofactors of a zero determinant in Section
2.3.7, it is seen that if A =0, r> 1, then J = 0. The right-hand side of the
above identity is also zero. Hence, in this particular case, the theorem is
valid but trivial. When r = 1, the theorem degenerates into the definition
and is again trivial. It therefore remains to prove the theorem
of A p 1 q 1
when A =0, r> 1.
The proof proceeds in two stages. In the first stage, the theorem is proved
in the particular case in which
p s = q s = s, 1 ≤ s ≤ r.