Determinants and Their Applications in Mathematical Physics

60 4. Particular Determinants

where

qij=

i

∑

r=1

U

(i)

ir

Pr(xj)

=

i

∑

r=1

U

(i)

ir

r→i

∑

s=1

asrx

s− 1

j

(asr=0, s>r)

=

i

∑

s=1

x

s− 1

j

i

∑

r=1

asrU

(i)

ir

=Ui

i

∑

s=1

x

s− 1

j δsi

=Uix

i− 1

j

Hence, referring to (4.1.8),

|qij|n=

(

U 1 U 2 ···Un)|x

i− 1

j

|

=Un|U

(i)

ij|nXn.

The theorem follows from (4.1.7) and (4.1.9).

4.1.7 A Generalized Vandermondian............

Lemma.
∣
∣
∣
∣
∣

N

∑

k=1

ykx

i+j− 2

k

∣ ∣ ∣ ∣ ∣ n

=

N

∑

k 1 ...kn=1

(

n

∏

r=1

ykr

)(

n

∏

s=2

x

s− 1

ks

)

∣

∣

x

i− 1

kj

∣

∣

n

Proof. Denote the determinant on the left byAnand put

a

(k)

ij

=ykx

i+j− 2

k

in the last identity in Property (g) in Section 2.3.1. Then,

An=

N

∑

k 1 ...kn=1

∣

∣y

kjx

i+j− 2

kj

∣

∣

n

Now remove the factoryk
j
x

j− 1

kj

from columnjof the determinant, 1≤j≤

n. The lemma then appears and is applied in Section 6.10.4 on the Einstein

and Ernst equations.

4.1.8 Simple Vandermondian Identities

Lemmas.

a.Vn=Vn− 1

n− 1

∏

r=1

(xn−xr),n> 1 ,V(x 1 )=1