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Circulant Matrix -- from Wolfram MathWorld

  • ️Weisstein, Eric W.
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An n×n matrix whose rows are composed of cyclically shifted versions of a length-n list l. For example, the 4×4 circulant matrix on the list l={1,2,3,4} is given by

C=[4 1 2 3; 3 4 1 2; 2 3 4 1; 1 2 3 4].

(1)

Circulant matrices are very useful in digital image processing, and the n×n 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 l and the second pads with zeros to give an n×n 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],

(2)

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

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

(3)

where omega_0=1, omega_1, ..., omega_(n-1) are the 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 when n=0 (mod 6).

Circulant matrices are examples of Latin squares.

See also

Circulant Determinant

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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.

Referenced on Wolfram|Alpha

Circulant Matrix

Cite this as:

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

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