Circulant matrix – Wikipedia
Linear algebra matrix
In linear algebra, a circulant matrix is a square matrix in which all row vectors are composed of the same elements and each row vector is rotated one element to the right relative to the preceding row vector. 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[edit]
An
circulant matrix
takes the form
or the transpose of this form (by choice of notation). When the term
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 0 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 matrix
.
Properties[edit]
Eigenvectors and eigenvalues[edit]
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[edit]
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[edit]
The rank of a circulant matrix
is equal to
, where
is the degree of the polynomial
.[2]
Other properties[edit]
- Any circulant is a matrix polynomial (namely, the associated polynomial) in the cyclic permutation matrix :
where
is given by
- 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 matrix 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.[3] Conversely, for any diagonal matrix
, the product
is circulant.
- Let be the (monic) characteristic polynomial of an circulant matrix , and let be the derivative of . Then the polynomial is the characteristic polynomial of the following submatrix of :
(see [4] for the proof).
Analytic interpretation[edit]
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
(
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[edit]
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
.
Symmetric circulant matrices belong to the class of bisymmetric matrices.
Hermitian circulant matrices[edit]
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[edit]
In linear equations[edit]
Given a matrix equation
where
is a circulant square 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[edit]
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[edit]
- ^ Davis, Philip J., Circulant Matrices, Wiley, New York, 1970 ISBN 0471057711
- ^ 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.
- ^ 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
- ^ 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
- ^ Tee, G J (2007). “Eigenvectors of Block Circulant and Alternating Circulant Matrices”. New Zealand Journal of Mathematics. 36: 195–211.
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