Page 66 - Determinants and Their Applications in Mathematical Physics

4

Particular Determinants

4.1 Alternants

4.1.1 Introduction

Any function of n variables which changes sign when any two of the vari-
ables are interchanged is known as an alternating function. It follows that
an alternating function vanishes if any two of the variables are equal.
Any determinant function which possess these properties is known as an
alternant.
The simplest form of alternant is

f 1 (x 1 ) f 2 (x 1 ) ··· f n (x 1 )

f 1 (x 2 ) f 2 (x 2 ) ···

. (4.1.1)
f n (x 2 )
|f j (x i )| n =
............................
f 1 (x n ) f 2 (x n ) ··· f n (x n )

n
The interchange of any two x’s is equivalent to the interchange of two rows
which gives rise to a change of sign. If any two of the x’s are equal, the
determinant has two identical rows and therefore vanishes.
The double or two-way alternant is
f(x 1 ,y 1 ) f(x 1 ,y 2 ) ··· f(x 1 ,y n )

f(x 2 ,y 1 ) f(x 2 ,y 2 ) ··· f(x 2 ,y n )

. (4.1.2)
|f(x i ,y j )| n =
...................................
f(x n ,y 1 ) f(x n ,y 2 ) ··· f(x n ,y n )

n