Page 69 - Determinants and Their Applications in Mathematical Physics

54 4. Particular Determinants

Postmultiply the Vandermondian V n (x)or V n (x 1 ,x 2 ,...,x n )by A n ,
prove the reduction formula

n

V n (x 1 ,x 2 ,...,x n )= V n−1 (x 2 − x 1 ,x 3 − x 1 ,...,x n − x 1 ) (x p − x 1 ),
p=2
and hence evaluate V n (x).
2. Prove that

y

|x j−1 n−j | n = r x r .
y
i i y s x s
1≤r<s≤n
3. If
x i = z + c i ,
ρ
prove that
j−1 −n(n−1)/2 j−1
|x | n = ρ |c | n ,
i i
which is independent of z. This relation is applied in Section 6.10.3 on
the Einstein and Ernst equations.

4.1.3 Cofactors of the Vandermondian
Theorem 4.1. The scaled cofactors of the Vandermonian X n = |x ij | n ,
j−1
where x ij = x are given by the quotient formula
i
(n)
(−1) n−j σ
X ij = i,n−j ,
g ni (x i )
n
where
n−1

g nr (x)= (−1) σ x .
s (n) n−1−s
rs
s=0
(n)
Notes on the symmetric polynomials σ rs and the function g nr (x) are given
in Appendix A.7.
Proof. Denote the quotient by F ij . Then,
1
n n
n−k (n) k−1
x ik F jk = (−1) σ x (Put k = n − s)
g nj (x j ) j,n−k i
k=1 k=1
n−1
1 s (n) n−s−1
= (−1) σ x
g nj (x j ) js i
s=0
g nj (x i )
=
g nj (x j )
= δ ij .