Page 75 - Determinants and Their Applications in Mathematical Physics

60 4. Particular Determinants

where
i
(i)
q ij = U P r (x j )
ir
r=1
i r→i
(i) s−1
= U a sr x (a sr =0, s>r)
ir j
r=1 s=1
i i
s−1 (i)
= x a sr U
j ir
s=1 r=1
i
s−1
x
= U i
j δ si
s=1
= U i x i−1 .
j
Hence, referring to (4.1.8),
i−1
|q ij | n = U 1 U 2 ··· U n )|x |
j
(i)
= U n |U | n X n .
ij
The theorem follows from (4.1.7) and (4.1.9).
4.1.7 A Generalized Vandermondian
Lemma.

N N n n

y k x = x i−1 .
x
i+j−2 s−1
y k r k s k j
k
n

k=1 k 1 ...k n =1 r=1 s=2
n
Proof. Denote the determinant on the left by A n and put
(k) i+j−2
a = y k x
ij
k
in the last identity in Property (g) in Section 2.3.1. Then,
N

i+j−2
A n = y k j x .
k j
n
k 1 ...k n =1
j−1
x from column j of the determinant, 1 ≤ j ≤
Now remove the factor y k j
k 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.
n−1

a. V n = V n−1 (x n − x r ), n > 1, V (x 1 )=1
r=1