Circulant matrix

Summary

In linear algebra, a circulant matrix is a square matrix in which all rows are composed of the same elements and each row is rotated one element to the right relative to the preceding row. It is a particular kind of Toeplitz matrix.

In numerical analysis, circulant matrices are important because they are diagonalized by a discrete Fourier transform, and hence linear equations that contain them may be quickly solved using a fast Fourier transform.[1] They can be interpreted analytically as the integral kernel of a convolution operator on the cyclic group and hence frequently appear in formal descriptions of spatially invariant linear operations. This property is also critical in modern software defined radios, which utilize Orthogonal Frequency Division Multiplexing to spread the symbols (bits) using a cyclic prefix. This enables the channel to be represented by a circulant matrix, simplifying channel equalization in the frequency domain.

In cryptography, a circulant matrix is used in the MixColumns step of the Advanced Encryption Standard.

Definition

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An   circulant matrix   takes the form   or the transpose of this form (by choice of notation). If each   is a   square matrix, then the   matrix   is called a block-circulant matrix.

A circulant matrix is fully specified by one vector,  , which appears as the first column (or row) of  . The remaining columns (and rows, resp.) of   are each cyclic permutations of the vector   with offset equal to the column (or row, resp.) index, if lines are indexed from   to  . (Cyclic permutation of rows has the same effect as cyclic permutation of columns.) The last row of   is the vector   shifted by one in reverse.

Different sources define the circulant matrix in different ways, for example as above, or with the vector   corresponding to the first row rather than the first column of the matrix; and possibly with a different direction of shift (which is sometimes called an anti-circulant matrix).

The polynomial   is called the associated polynomial of the matrix  .

Properties

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Eigenvectors and eigenvalues

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The normalized eigenvectors of a circulant matrix are the Fourier modes, namely,   where   is a primitive  -th root of unity and   is the imaginary unit.

(This can be understood by realizing that multiplication with a circulant matrix implements a convolution. In Fourier space, convolutions become multiplication. Hence the product of a circulant matrix with a Fourier mode yields a multiple of that Fourier mode, i.e. it is an eigenvector.)

The corresponding eigenvalues are given by  

Determinant

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As a consequence of the explicit formula for the eigenvalues above, the determinant of a circulant matrix can be computed as:   Since taking the transpose does not change the eigenvalues of a matrix, an equivalent formulation is  

Rank

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The rank of a circulant matrix   is equal to   where   is the degree of the polynomial  .[2]

Other properties

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  • Any circulant is a matrix polynomial (namely, the associated polynomial) in the cyclic permutation matrix  :   where   is given by the companion matrix  
  • The set of   circulant matrices forms an  -dimensional vector space with respect to addition and scalar multiplication. This space can be interpreted as the space of functions on the cyclic group of order  ,  , or equivalently as the group ring of  .
  • Circulant matrices form a commutative algebra, since for any two given circulant matrices   and  , the sum   is circulant, the product   is circulant, and  .
  • For a nonsingular circulant matrix  , its inverse   is also circulant. For a singular circulant matrix, its Moore–Penrose pseudoinverse   is circulant.
  • The discrete Fourier transform matrix of order   is defined as by

  There are important connections between circulant matrices and the DFT matrices. In fact, it can be shown that   where   is the first column of  . The eigenvalues of   are given by the product  . This product can be readily calculated by a fast Fourier transform.[3]

  • Let   be the (monic) characteristic polynomial of an   circulant matrix  . Then the scaled derivative   is the characteristic polynomial of the following   submatrix of  :   (see [4] for the proof).

Analytic interpretation

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Circulant matrices can be interpreted geometrically, which explains the connection with the discrete Fourier transform.

Consider vectors in   as functions on the integers with period  , (i.e., as periodic bi-infinite sequences:  ) or equivalently, as functions on the cyclic group of order   (denoted   or  ) geometrically, on (the vertices of) the regular  -gon: this is a discrete analog to periodic functions on the real line or circle.

Then, from the perspective of operator theory, a circulant matrix is the kernel of a discrete integral transform, namely the convolution operator for the function  ; this is a discrete circular convolution. The formula for the convolution of the functions   is

 

(recall that the sequences are periodic) which is the product of the vector   by the circulant matrix for  .

The discrete Fourier transform then converts convolution into multiplication, which in the matrix setting corresponds to diagonalization.

The  -algebra of all circulant matrices with complex entries is isomorphic to the group  -algebra of  

Symmetric circulant matrices

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For a symmetric circulant matrix   one has the extra condition that  . Thus it is determined by   elements.  

The eigenvalues of any real symmetric matrix are real. The corresponding eigenvalues   become:   for   even, and   for   odd, where   denotes the real part of  . This can be further simplified by using the fact that   and   depending on   even or odd.

Symmetric circulant matrices belong to the class of bisymmetric matrices.

Hermitian circulant matrices

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The complex version of the circulant matrix, ubiquitous in communications theory, is usually Hermitian. In this case   and its determinant and all eigenvalues are real.

If n is even the first two rows necessarily takes the form   in which the first element   in the top second half-row is real.

If n is odd we get  

Tee[5] has discussed constraints on the eigenvalues for the Hermitian condition.

Applications

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In linear equations

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Given a matrix equation

 

where   is a circulant matrix of size  , we can write the equation as the circular convolution   where   is the first column of  , and the vectors  ,   and   are cyclically extended in each direction. Using the circular convolution theorem, we can use the discrete Fourier transform to transform the cyclic convolution into component-wise multiplication   so that  

This algorithm is much faster than the standard Gaussian elimination, especially if a fast Fourier transform is used.

In graph theory

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In graph theory, a graph or digraph whose adjacency matrix is circulant is called a circulant graph/digraph. Equivalently, a graph is circulant if its automorphism group contains a full-length cycle. The Möbius ladders are examples of circulant graphs, as are the Paley graphs for fields of prime order.

References

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  1. ^ Davis, Philip J (1970). Circulant Matrices. New York: Wiley. ISBN 0-471-05771-1. OCLC 1408988930.
  2. ^ A. W. Ingleton (1956). "The Rank of Circulant Matrices". J. London Math. Soc. s1-31 (4): 445–460. doi:10.1112/jlms/s1-31.4.445.
  3. ^ Golub, Gene H.; Van Loan, Charles F. (1996), "§4.7.7 Circulant Systems", Matrix Computations (3rd ed.), Johns Hopkins, ISBN 978-0-8018-5414-9
  4. ^ Kushel, Olga; Tyaglov, Mikhail (July 15, 2016), "Circulants and critical points of polynomials", Journal of Mathematical Analysis and Applications, 439 (2): 634–650, arXiv:1512.07983, doi:10.1016/j.jmaa.2016.03.005, ISSN 0022-247X
  5. ^ Tee, G.J. (2007). "Eigenvectors of Block Circulant and Alternating Circulant Matrices" (PDF). New Zealand Journal of Mathematics. 36: 195–211.
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  • Gray, R.M. (2006). Toeplitz and Circulant Matrices: A Review (PDF). Now. doi:10.1561/0100000006. ISBN 978-1-933019-68-0 – via Stanford University.
  • IPython Notebook demonstrating properties of circulant matrices