Determinants and Their Applications in Mathematical Physics

4.4 Circulants 79

that is, the Pfaffian is equal to the product of its elements.

4.4 Circulants............................

4.4.1 Definition and Notation................

A circulantAnis denoted by the symbolA(a 1 ,a 2 ,a 3 ,...,an) and is defined

as follows:

An=A(a 1 ,a 2 ,a 3 ,...,an)

=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

∣

a 1 a 2 a 3 ··· an

an a 1 a 2 ··· an− 1

an− 1 an a 1 ··· an− 2

··· ··· ··· ··· ···

a 2 a 3 a 4 ··· a 1

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n

. (4.4.1)

Each row is obtained from the previous row by displacing each element,

except the last, one position to the right, the last element being displaced

to the first position. The name circulant is derived from the circular nature

of the displacements.

An=|aij|n,

where

aij=

{

aj+1−i,j≥i,

an+j+1−i,j<i.

(4.4.2)

4.4.2 Factors

After performing the column operation

C

′

1

=

n

∑

j=1

Cj, (4.4.3)

it is easily seen thatAnhas the factor

∑n

r=1

arbutAnhas other factors.

When all thearare real, the first factor is real but some of the other

factors are complex.

Letωrdenote the complex number defined as follows and let ̄ωrdenote

its conjugate:

ωr= exp(2riπ/n)0≤r≤n− 1 ,

=ω

r

1

,

ω

n

r

=1,

ωrω ̄r=1,

ω 0 =1. (4.4.4)