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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 circulant A n is denoted by the symbol A(a 1 ,a 2 ,a 3 ,...,a n ) and is defined
as follows:
A n = A(a 1 ,a 2 ,a 3 ,...,a n )

a 1 a 2 a 3 ··· a n

a 1 a 2 ··· a n−1
a n

= a n−1 a n a 1 ··· a n−2 . (4.4.1)

··· ··· ··· ··· ···

a 2 a 3 a 4 ··· a 1
n
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.
A n = |a ij | n ,
where
a j+1−i , j ≥ i,

a ij = (4.4.2)
a n+j+1−i ,j < i.
4.4.2 Factors

After performing the column operation
n

C = C j , (4.4.3)
1
j=1
it is easily seen that A n has the factor n ! a r but A n has other factors.
r=1
When all the a r are real, the first factor is real but some of the other
factors are complex.
Let ω r denote the complex number defined as follows and let ¯ω r denote
its conjugate:
ω r = exp(2riπ/n)0 ≤ r ≤ n − 1,
= ω ,
r
1
ω =1,
n
r
ω r ¯ω r =1,
ω 0 =1. (4.4.4)