Determinants and Their Applications in Mathematical Physics

54 4. Particular Determinants

Postmultiply the VandermondianVn(x)orVn(x 1 ,x 2 ,...,xn)byAn,

prove the reduction formula

Vn(x 1 ,x 2 ,...,xn)=Vn− 1 (x 2 −x 1 ,x 3 −x 1 ,...,xn−x 1 )

n

∏

p=2

(xp−x 1 ),

and hence evaluateVn(x).

2.Prove that

|x

j− 1

i y

n−j

i |n=

∏

1 ≤r<s≤n

∣

∣

∣

∣

yr xr

ys xs

∣

∣

∣

∣

3.If

xi=

z+ci

ρ

,

prove that

|x

j− 1

i

|n=ρ

−n(n−1)/ 2

|c

j− 1

i

|n,

which is independent ofz. 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 VandermonianXn=|xij|n,

wherexij=x

j− 1

i

are given by the quotient formula

X

ij

n=

(−1)

n−j

σ

(n)

i,n−j

gni(xi)

,

where

gnr(x)=

n− 1

∑

s=0

(−1)

s

σ

(n)

rs

x

n− 1 −s

.

Notes on the symmetric polynomialsσ

(n)

rs and the functiongnr(x)are given

in Appendix A.7.

Proof. Denote the quotient byFij. Then,

n

∑

k=1

xikFjk=

1

gnj(xj)

n

∑

k=1

(−1)

n−k

σ

(n)

j,n−k

x

k− 1

i

(Putk=n−s)

=

1

gnj(xj)

n− 1

∑

s=0

(−1)

s

σ

(n)

js

x

n−s− 1

i

=

gnj(xi)

gnj(xj)

=δij.