Determinants and Their Applications in Mathematical Physics

100 4. Particular Determinants

= 0 if the parameters are not distinct

W

′

ijk...r

= the sum of the determinants obtained by increasing

the parameters one at a time by 1 and discarding

those determinants with two identical parameters. (4.7.8)

Illustration.Let

W=

∣

∣

CC

′

C

′′

∣

∣

=W 012.

Then

W

′

=W 013 ,

W

′′

=W 014 +W 023 ,

W

′′′

=W 015 +2W 024 +W 123 ,

W

(4)

=W 016 +3W 025 +2W 034 +3W 124 ,

W

(5)

=W 017 +4W 026 +5W 035 +6W 125 +5W 134 , (4.7.9)

etc. Formulas of this type appear in Sections 6.7 and 6.8 on the K dV and

KP equations.

4.7.3 The Derivative of a Cofactor.............

In order to determine formulas for (W

(n)

ij

)

′

, it is convenient to change the

notation used in the previous section.

Let

W=|wij|n,

where

wij=y

(j−1)

i

=D

j− 1

(yi),D=

d

dx

,

and where theyiare arbitrary (n−1) differentiable functions.

Clearly,

w

′

ij

=wi,j+1.

In column vector notation,

Wn=

∣

∣C

1 C 2 ···Cn

∣

∣,

where

Cj=

[

y

(j−1)

1 y

(j−1)

2 ···y

(j−1)

n

]T

,

C

′

j=Cj+1.

Theorem 4.26.

a.

(

W

(n)

ij

)′

=−W

(n)

i,j− 1

−W

(n+1)

i,n+1;jn

.

b.

(

W

(n)

i 1

)′

=−W

(n+1)

i,n+1;1n

.