Page 77 - Determinants and Their Applications in Mathematical Physics

62 4. Particular Determinants

referring to Lemma (a) with n → s + 1 and Lemma (c) with m → s +1,
 
s+1 s−1
 V (x s+1 ,x s+2 ,...,x n )V

M r =  V n  n
 V (x 1 ,x 2 ,...,x s )
n
 s
r=1 (x s+1 − x r ) (x r − x s+1 )
r=1 r=s+2
 
V s  V (x s+1 ,x s+2 ,...,x n ) 
= n  
s  n 
V (x 1 ,x 2 ,...,x s ) (x s+1 − x r ) (x r − x s+1 )
r=1 r=s+2
V (x s+2 ,x s+3 ,...,x n )V s
= n .
V (x 1 ,x 2 ,...,x s+1 )
Hence, (a) is valid when m = s + 1, which proves (a). To prove (b), put
m = n − 1 in (a) and use M n = V n−1 . The details are elementary.
The proof of (c) is obtained by applying the permutation

1 2 3 ··· n
k 1 k 2 k 3 ··· k n
to (a). The only complication which arises is the determination of the sign
of the expression on the right of (c). It is left as an exercise for the reader
to prove that the sign is positive.
Exercise. Let A 6 denote the determinant of order 6 defined in column
vector notation as follows:

2
C j = a j a j x j a j x b j b j y j b j y 2 T , 1 ≤ j ≤ 6.
j j
Apply the Laplace expansion theorem to prove that

A 6 = σa i a j a k b p b q b r V (x i ,x j ,x k )V (y p ,y q ,y r ),
i<j<k
p<q<r
where

1 2 3 4 5 6
σ = sgn
i j k p q r
and where the lower set of parameters is a permutation of the upper set.
6

The number of terms in the sum is = 20.
3
Prove also that
A 6 = 0 when a j = b j , 1 ≤ j ≤ 6.
Generalize this result by giving an expansion formula for A 2n from the
first m rows and the remaining (2n − m) rows using the dummy variables
k r ,1 ≤ r ≤ 2n. The generalized formula and Theorem (c) are applied in
Section 6.10.4 on the Einstein and Ernst equations.