Determinants and Their Applications in Mathematical Physics

4.3 Skew-Symmetric Determinants 65

=

∣

∣

∣

∣

Apr Aps

Aqr Aqs

∣

∣

∣

∣

+

∣

∣

∣

∣

Apq Aqs

Apr Ars

∣

∣

∣

∣

+

∣

∣

∣

∣

Aqr Ars

Apq Aps

∣

∣

∣

∣

=0.

The theorem follows immediately ifA= 0. However, since the identity is

purely algebraic, all the terms in the expansion ofSas 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 whichA= 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 determinantAn=|aij|nin whichaji=−aij, which impliesaii=0,

is said to be skew-symmetric. In detail,

An=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

• a 12 a 13 a 14 ...

−a 12 • a 23 a 24 ...

−a 13 −a 23 • a 34 ...

−a 14 −a 24 −a 34 • ...

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

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n

. (4.3.1)

Theorem 4.8. The square of an arbitrary determinant of orderncan be

expressed as a symmetric determinant of ordernifnis odd or a skew-

symmetric determinant of ordernifnis even.

Proof. Let

A=|aij|n.

Reversing the order of the rows,

A=(−1)

N

|an+1−i,j|n,N=

[

n

2

]

. (4.3.2)

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)

N

|(−1)

i+1

an+1−j,n+1−i|n. (4.3.3)

Hence, applying the formula for the product of two determinants in

Section 1.4,

A

2

=|an+1−i,j|n|(−1)

i+1

an+1−j,n+1−i|n