Circulant matrix explained
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.
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
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).
is called the
associated polynomial of the matrix
.
Properties
Eigenvectors and eigenvalues
The normalized eigenvectors of a circulant matrix are the Fourier modes, namely,where
\omega=\exp\left(\tfrac{2\pii}{n}\right)
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
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
The rank of a circulant matrix
is equal to
where
is the
degree of the polynomial
.
[2] Other properties
:
where
is given by the
companion matrix
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
.
and
, the sum
is circulant, the product
is circulant, and
.
, its
inverse
is also circulant. For a singular circulant matrix, its
Moore–Penrose pseudoinverse
is circulant.
that is composed of the eigenvectors of a circulant matrix is related to the discrete Fourier transform and its inverse transform:
Consequently the matrix
diagonalizes
. In fact, we have
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. Conversely, for any diagonal matrix
, the product
is circulant.
be the (
monic)
characteristic polynomial of an
circulant matrix
. Then the scaled
derivative is the characteristic polynomial of the following
submatrix of
:
(see for the
proof).
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
, (i.e., as periodic bi-infinite sequences:
...,a0,a1,...,an-1,a0,a1,...
) or equivalently, as functions on the
cyclic group of order
(denoted
or
) geometrically, on (the vertices of) the
regular : 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
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
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 formin which the first element
in the top second half-row is real.
If n is odd we get
Tee[3] has discussed constraints on the eigenvalues for the Hermitian condition.
Applications
In linear equations
Given a matrix equation
where
is a circulant matrix of size
, we can write the equation as the
circular convolutionwhere
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
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.
External links
Notes and References
- [Philip J. Davis|Davis, Philip J.]
- A. W. Ingleton . The Rank of Circulant Matrices . J. London Math. Soc. . 1956 . s1-31 . 4 . 445–460 . 10.1112/jlms/s1-31.4.445.
- Tee. G J. 2007. Eigenvectors of Block Circulant and Alternating Circulant Matrices. New Zealand Journal of Mathematics. 36. 195–211.