Circulant matrix

In linear algebra, a circulant matrix is a special kind of Toeplitz matrix where each row vector is rotated one element to the right relative to the preceding row vector. 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. They can be interpreted analytically as the integral kernel of a convolution operator on the cyclic group In cryptography, a circulant matrix is used in the MixColumns step of the Advanced Encryption Standard.

Definition

An matrix of the form

is called a circulant matrix.

A circulant matrix is fully specified by one vector, , which appears as the first column of . The remaining columns of are each cyclic permutations of the vector with offset equal to the column index. The last row of is the vector in reverse order, and the remaining rows are each cyclic permutations of the last row.

Properties

The set of circulant matrices forms an n-dimensional vector space; this can be interpreted as the space of functions on the cyclic group of order n, or equivalently the group ring.

Circulant matrices form a commutative algebra, since for any two given circulant matrices and , the sum is circulant, the product is circulant, and .

The eigenvectors of a circulant matrix of a given size are the columns of the discrete Fourier transform matrix of the same size. The latter matrix is defined by

Thus, the matrix F_n diagonalizes C. In fact, we have

where is the first column of . Thus, the eigenvalues of C are given by the product . This product can be readily calculated by a Fast Fourier transform (Golub & van Loan 1996, §4.7.7).

Solving linear equations with circulant matrices

Given a matrix equation

where is a circulant square matrix of size we can write the equation as the cyclic convolution

where is the first column of , and the vectors , and are cyclically extended in each direction. Using the results of 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.

Analytic interpretation

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 n, (i.e., as periodic bi-infinite sequences: ) or equivalently, as functions on the cyclic group of order n, (C_n or ) geometrically, on (the vertices of) the regular n-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 (b_i): = (c_i) * (a_i) is

which is the product of the vector of a_i by the circulant matrix.

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

Application in graph theory

In graph theory, a graph or digraph whose adjacency matrix is circulant is called a circulant graph (or 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

• Golub, Gene H.; Van Loan, Charles F. (1996), Matrix Computations (3rd ed.), Johns Hopkins, ISBN 978-0-8018-5414-9 .

External links

• Toeplitz and Circulant Matrices: A Review, by R. M. Gray