Page 115 - Determinants and Their Applications in Mathematical Physics

100 4. Particular Determinants

= 0 if the parameters are not distinct
W = the sum of the determinants obtained by increasing
ijk...r
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)
) , it is convenient to change the
ij
notation used in the previous section.
Let
W = |w ij | n ,
where
d
(j−1) j−1
w ij = y = D (y i ), D = ,
dx
i
and where the y i are arbitrary (n − 1) differentiable functions.
Clearly,

w = w i,j+1 .
ij
In column vector notation,

W n = C 1 C 2 ··· C n ,

where
(j−1) (j−1)
C j = y y ··· y (j−1) T ,
1 2 n
C = C j+1 .

j
Theorem 4.26.
(n) (n) (n+1)

a. W = −W i,j−1 − W .
ij i,n+1;jn
(n) (n+1)

b. W = −W .
i1 i,n+1;1n