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Circulant Matrix

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A circulant matrix is an n×n matrix in which each row after the first is obtained by cyclically shifting the preceding row one position to the right, where n is a positive integer. For example, beginning with a right shift of the list l={1,2,3,4} gives

C=[4 1 2 3; 3 4 1 2; 2 3 4 1; 1 2 3 4].
(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 l is a nonempty vector. To obtain an n×n matrix with zeros padded on the left, replace l by PadLeft[l, n], where n is a positive integer at least the length of l.

A special circulant matrix, for a positive integer n, 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],,
(2)

where (n; k) is a binomial coefficient. The determinant of C_n is given by

det(C_n)=product_(j=0)^(n-1)[(1+omega_j)^n-1],
(3)

where omega_j=e^(2piij/n) for j=0, 1, ..., n-1 are the distinct nth roots of unity. The determinants for n=1, 2, ..., are given by 1, -3, 28, -375, 3751, 0, 6835648, -1343091375, 364668913756, ... (OEIS A048954), which is 0 iff n=0 (mod 6).

A circulant matrix is a Latin square iff its first row has n distinct entries.

See also

Circulant Determinant, Convolution Matrix, Toeplitz Matrix

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References

Davis, P. J. Circulant Matrices, 2nd ed. New York: Chelsea, 1994.Sloane, N. J. A. Sequence A048954 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.

Referenced on Wolfram|Alpha

Circulant Matrix

Cite this as:

Weisstein, Eric W. "Circulant Matrix." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CirculantMatrix.html

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