Page 67 - Determinants and Their Applications in Mathematical Physics

52 4. Particular Determinants

If the x’s are not distinct, the determinant has two or more identical rows. If
the y’s are not distinct, the determinant has two or more identical columns.
In both cases, the determinant vanishes.
Illustration. The Wronskian |D j−1 (f i )| n is an alternant. The double
x
Wronskian |D j−1 D i−1 (f)| n is a double alternant, D x = ∂/∂x, etc.
x y
Exercise. Define two third-order alternants φ and ψ in column vector
notation as follows:

φ = |c(x 1 ) c(x 2 ) c(x 3 )|,
ψ = |C(x 1 ) C(x 2 ) C(x 3 )|.

Apply l’Hopital’s formula to prove that

φ |c(x) c (x) c (x)|

lim = ,
ψ |C(x) C (x) C (x)|

where the limit is carried out as x i → x,1 ≤ i ≤ 3, provided the numerator
and denominator are not both zero.
4.1.2 Vandermondians
The determinant
j−1
X n = |x
i | n
1 x 1 x 2 1 ··· x n−1

1
1 x 2 x 2 2 ··· x n−1

2

=
......................
1 x ··· x
2 n−1
x n
n n
n
= V (x 1 ,x 2 ,...,x n ) (4.1.3)
is known as the alternant of Vandermonde or simply a Vandermondian.
Theorem.

X n = (x s − x r ).
1≤r<s≤n
The expression on the right is known as a difference–product and contains
1
(n/2) = n(n − 1) factors.
2
First Proof. The expansion of the determinant consists of the sum of n!
terms, each of which is the product of n elements, one from each row and
one from each column. Hence, X n is a polynomial in the x r of degree
1
0+1+2+3+ ··· +(n − 1) = n(n − 1).
2
One of the terms in this polynomial is the product of the elements in the
leading diagonal, namely
2 3
+ x 2 x x ··· x n−1 . (4.1.4)
3 4 n