An 
matrix whose rows are composed of cyclically shifted versions
of a length-
list 
.
For example, the 
circulant matrix on the list 
is given by
![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 very useful in digital image processing, and the 
circulant matrix is implemented as CirculantMatrix[l,
n] in the Mathematica application
package Digital
Image Processing.
Circulant matrices can be implemented in the Wolfram Language as follows.
CirculantMatrix[l_List?VectorQ] :=
NestList[RotateRight, RotateRight[l],
Length[l] - 1]
CirculantMatrix[l_List?VectorQ, n_Integer] :=
NestList[RotateRight,
RotateRight[Join[Table[0, {n - Length[l]}],
l]], n - 1] /; n >= Length[l]
where the first input creates a matrix with dimensions equal to the length of 
and the second pads with zeros to give
an 
matrix. A special type of circulant matrix 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 the beautiful formula
![C_n=product_(j=0)^(n-1)[(1+omega_j)^n-1],](/images/equations/CirculantMatrix/NumberedEquation3.svg) |
(3)
|
where 
,
, ..., 
are the 
th roots of unity. The determinants
for 
,
2, ..., are given by 1, 
, 28, 
, 3751, 0, 6835648, 
, 364668913756, ... (OEIS A048954),
which is 0 when 
.
Circulant matrices are examples of Latin squares.
