Page 80 - Determinants and Their Applications in Mathematical Physics

4.3 Skew-Symmetric Determinants 65

= A pr A ps + A pq A qs + A qr A rs
A qr A qs A pr A rs A pq A ps
=0.
The theorem follows immediately if A = 0. However, since the identity is
purely algebraic, all the terms in the expansion of S as sums of products
of elements must cancel out in pairs. The identity must therefore be valid
for all values of its elements, including those values for which A = 0. The
theorem is clearly valid if the sum is carried out over even permutations of
any three of the four parameters.
Notes on skew-symmetric, circulant, centrosymmetric, skew-centrosym-
metric, persymmetric (Hankel) determinants, and symmetric Toeplitz
determinants are given under separate headings.

4.3 Skew-Symmetric Determinants

4.3.1 Introduction
The determinant A n = |a ij | n in which a ji = −a ij , which implies a ii =0,
is said to be skew-symmetric. In detail,

• a 12 a 13 a 14 ...

−a 12 • a 23 a 24 ...

−a 13 −a 23 • a 34 ...
. (4.3.1)
A n =
−a 14 −a 24 −a 34 • ...

.............................

n
Theorem 4.8. The square of an arbitrary determinant of order n can be
expressed as a symmetric determinant of order n if n is odd or a skew-
symmetric determinant of order n if n is even.
Proof. Let
A = |a ij | n .
Reversing the order of the rows,
n

A =(−1) |a n+1−i,j | n , N = . (4.3.2)
N
2
Transposing the elements of the original determinant across the secondary
diagonal and changing the signs of the elements in the new rows 2, 4, 6,...,
A =(−1) |(−1) i+1 a n+1−j,n+1−i | n . (4.3.3)
N
Hence, applying the formula for the product of two determinants in
Section 1.4,
2
A = |a n+1−i,j | n |(−1) i+1 a n+1−j,n+1−i | n