A circulant matrix is an 
matrix in which each row
after the first is obtained by cyclically shifting the preceding row one position
to the right, where 
is a positive integer. For example, beginning
with a right shift of the list 
gives
![C=[4 1 2 3; 3 4 1 2; 2 3 4 1; 1 2 3 4].](/images/equations/CirculantMatrix/NumberedEquation1.svg) |
(1)
|
Circulant matrices are Toeplitz matrices and are useful in digital image processing. In the Wolfram
Language, the convention above is obtained using ToeplitzMatrix[Reverse[l],
RotateRight[l]], where 
is a nonempty vector. To obtain
an 
matrix with zeros padded on the left, replace 
by PadLeft[l,
n], where 
is a positive integer at least the length of
.
A special circulant matrix, for a positive integer 
, is defined as
![C_n=[1 (n; 1) (n; 2) ... (n; n-1); (n; n-1) 1 (n; 1) ... (n; n-2); | | | ... |; (n; 1) (n; 2) (n; 3) ... 1],,](/images/equations/CirculantMatrix/NumberedEquation2.svg) |
(2)
|
where 
is a binomial coefficient. The determinant
of 
is given by
![det(C_n)=product_(j=0)^(n-1)[(1+omega_j)^n-1],](/images/equations/CirculantMatrix/NumberedEquation3.svg) |
(3)
|
where 
for 
,
1, ..., 
are the distinct 
th
roots of unity. The determinants
for 
,
2, ..., are given by 1, 
, 28, 
, 3751, 0, 6835648, 
, 364668913756, ... (OEIS A048954),
which is 0 iff 
.
A circulant matrix is a Latin square iff its first row has 
distinct entries.
