Determinants and Their Applications in Mathematical Physics

38 3. Intermediate Determinant Theory

Hence, removing the factorAnfrom each row,

|cij|n=A

n

n

∣

∣

∣

∣

δijxi+

Hij

An

∣

∣

∣

∣

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 determinantA=|aij|n, the rejecter minorMp 1 p 2 ...pr;q 1 q 2 ...qr

of order (n−r) and the retainer minorNp 1 p 2 ...pr;q 1 q 2 ...qr of orderr are

defined in Section 3.2.1.

Define the retainer minorJof orderras follows:

J=Jp

1 p 2 ...pr;q 1 q 2 ...qr

= adjNp

1 p 2 ...pr;q 1 q 2 ...qr

=

∣ ∣ ∣ ∣ ∣ ∣ ∣

Ap 1 q 1 Ap 2 q 1 ··· Aprq 1

Ap 1 q 2 Ap 2 q 2 ··· Aprq 2

.........................

Ap

1 qr

Ap

2 qr

··· Ap

rqr

∣ ∣ ∣ ∣ ∣ ∣ ∣ r

. (3.6.1)

Jis a minor of adjA. For example,

J 23 , 24 = adjN 23 , 24

= adj

∣

∣

∣

∣

a 22 a 24

a 32 a 34

∣

∣

∣

∣

=

∣

∣

∣

∣

A 22 A 32

A 24 A 34

∣

∣

∣

∣

.

The Jacobi identity on the minors of adjAis given by the following theorem:

Theorem.

Jp

1 p 2 ...pr;q 1 q 2 ...qr

=A

r− 1

Mp

1 p 2 ...pr;q 1 q 2 ...qr

, 1 ≤r≤n− 1.

Referring to the section on the cofactors of a zero determinant in Section

2.3.7, it is seen that ifA=0,r>1, thenJ= 0. The right-hand side of the

above identity is also zero. Hence, in this particular case, the theorem is

valid but trivial. Whenr= 1, the theorem degenerates into the definition

ofAp 1 q 1 and is again trivial. It therefore remains to prove the theorem

whenA=0,r>1.

The proof proceeds in two stages. In the first stage, the theorem is proved

in the particular case in which

ps=qs=s, 1 ≤s≤r.