Circulant Matrix
المؤلف:
Davis, P. J.
المصدر:
Circulant Matrices, 2nd ed. New York: Chelsea, 1994.
الجزء والصفحة:
...
23-12-2021
2368
Circulant Matrix
An 
matrix whose rows are composed of cyclically shifted versions of a length-
list 
. For example, the 
circulant matrix on the list ![l=<span style=]()
{1,2,3,4}" src="https://mathworld.wolfram.com/images/equations/CirculantMatrix/Inline5.gif" style="height:16px; width:80px" /> is given by
![C=[4 1 2 3; 3 4 1 2; 2 3 4 1; 1 2 3 4].](https://mathworld.wolfram.com/images/equations/CirculantMatrix/NumberedEquation1.gif) |
(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],](https://mathworld.wolfram.com/images/equations/CirculantMatrix/NumberedEquation2.gif) |
(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],](https://mathworld.wolfram.com/images/equations/CirculantMatrix/NumberedEquation3.gif) |
(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.
REFERENCES:
Davis, P. J. Circulant Matrices, 2nd ed. New York: Chelsea, 1994.
Sloane, N. J. A. Sequences A048954 and A049287 in "The On-Line Encyclopedia of Integer Sequences."
Stroeker, R. J. "Brocard Points, Circulant Matrices, and Descartes' Folium." Math. Mag. 61, 172-187, 1988.
Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, p. 114, 1991.
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