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# Circulant matrix

In linear algebra, a circulant matrix is a square matrix in which 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 ${\displaystyle C_{n}}$ and hence frequently appear in formal descriptions of spatially invariant linear operations.

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

## Definition

An ${\displaystyle n\times n}$ circulant matrix ${\displaystyle C}$ takes the form

${\displaystyle C={\begin{bmatrix}c_{0}&c_{n-1}&\dots &c_{2}&c_{1}\\c_{1}&c_{0}&c_{n-1}&&c_{2}\\\vdots &c_{1}&c_{0}&\ddots &\vdots \\c_{n-2}&&\ddots &\ddots &c_{n-1}\\c_{n-1}&c_{n-2}&\dots &c_{1}&c_{0}\\\end{bmatrix}}}$

or the transpose of this form (by choice of notation).

A circulant matrix is fully specified by one vector, ${\displaystyle c}$, which appears as the first column (or row) of ${\displaystyle C}$. The remaining columns (and rows, resp.) of ${\displaystyle C}$ are each cyclic permutations of the vector ${\displaystyle c}$ with offset equal to the column (or row, resp.) index, if lines are indexed from 0 to ${\displaystyle n-1}$. (Cyclic permutation of rows has the same effect as cyclic permutation of columns.) The last row of ${\displaystyle C}$ is the vector ${\displaystyle c}$ shifted by one in reverse.

Different sources define the circulant matrix in different ways, for example as above, or with the vector ${\displaystyle c}$ 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 ${\displaystyle f(x)=c_{0}+c_{1}x+\dots +c_{n-1}x^{n-1}}$ is called the associated polynomial of matrix ${\displaystyle C}$.

## Properties

### Eigenvectors and eigenvalues

The normalized eigenvectors of a circulant matrix are the Fourier modes, namely,

${\displaystyle v_{j}={\frac {1}{\sqrt {n}}}(1,\omega ^{j},\omega ^{2j},\ldots ,\omega ^{(n-1)j}),\quad j=0,1,\ldots ,n-1,}$

where ${\displaystyle \omega =\exp \left({\tfrac {2\pi i}{n}}\right)}$ is a primitive ${\displaystyle n}$-th root of unity and ${\displaystyle i}$ is the imaginary unit.

(This can be understood by realizing that a circulant matrix implements a convolution.)

The corresponding eigenvalues are then given by

${\displaystyle \lambda _{j}=c_{0}+c_{n-1}\omega ^{j}+c_{n-2}\omega ^{2j}+\ldots +c_{1}\omega ^{(n-1)j},\quad j=0,1,\ldots ,n-1.}$

### Determinant

As a consequence of the explicit formula for the eigenvalues above, the determinant of a circulant matrix can be computed as:

${\displaystyle \det(C)=\prod _{j=0}^{n-1}(c_{0}+c_{n-1}\omega ^{j}+c_{n-2}\omega ^{2j}+\dots +c_{1}\omega ^{(n-1)j}).}$

Since taking the transpose does not change the eigenvalues of a matrix, an equivalent formulation is

${\displaystyle \det(C)=\prod _{j=0}^{n-1}(c_{0}+c_{1}\omega ^{j}+c_{2}\omega ^{2j}+\dots +c_{n-1}\omega ^{(n-1)j})=\prod _{j=0}^{n-1}f(\omega ^{j}).}$

### Rank

The rank of a circulant matrix ${\displaystyle C}$ is equal to ${\displaystyle n-d}$, where ${\displaystyle d}$ is the degree of the polynomial ${\displaystyle \gcd(f(x),x^{n}-1)}$.[2]

### Other properties

• Any circulant is a matrix polynomial (namely, the associated polynomial) in the cyclic permutation matrix ${\displaystyle P}$:
${\displaystyle C=c_{0}I+c_{1}P+c_{2}P^{2}+\ldots +c_{n-1}P^{n-1}=f(P),}$
where ${\displaystyle P}$ is given by
${\displaystyle P={\begin{bmatrix}0&0&\ldots &0&1\\1&0&\ldots &0&0\\0&\ddots &\ddots &\vdots &\vdots \\\vdots &\ddots &\ddots &0&0\\0&\ldots &0&1&0\end{bmatrix}}.}$
• The set of ${\displaystyle n\times n}$ circulant matrices forms an ${\displaystyle n}$-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 ${\displaystyle n}$, ${\displaystyle C_{n}}$, or equivalently as the group ring of ${\displaystyle C_{n}}$.
• Circulant matrices form a commutative algebra, since for any two given circulant matrices ${\displaystyle A}$ and ${\displaystyle B}$, the sum ${\displaystyle A+B}$ is circulant, the product ${\displaystyle AB}$ is circulant, and ${\displaystyle AB=BA}$.
• The matrix ${\displaystyle U}$ that is composed of the eigenvectors of a circulant matrix is related to the discrete Fourier transform and its inverse transform:
${\displaystyle U_{n}^{*}={\frac {1}{\sqrt {n}}}F_{n},\quad {\text{and}}\quad U_{n}={\frac {1}{\sqrt {n}}}F_{n}^{-1},{\text{ where }}F_{n}=(f_{jk}){\text{ with }}f_{jk}=e^{-2jk\pi i/n},\,{\text{for }}0\leq j,k
Consequently the matrix ${\displaystyle U_{n}}$ diagonalizes ${\displaystyle C}$. In fact, we have
${\displaystyle C=U_{n}\operatorname {diag} (F_{n}c)U_{n}^{*}={\frac {1}{n}}F_{n}^{-1}\operatorname {diag} (F_{n}c)F_{n},}$
where ${\displaystyle c}$ is the first column of ${\displaystyle C}$. The eigenvalues of ${\displaystyle C}$ are given by the product ${\displaystyle F_{n}c}$. This product can be readily calculated by a fast Fourier transform.[3]
• Let ${\displaystyle p(x)}$ be the (monic) characteristic polynomial of an ${\displaystyle n\times n}$ circulant matrix ${\displaystyle C}$, and let ${\displaystyle p'(x)}$ be the derivative of ${\displaystyle p(x)}$. Then the polynomial ${\displaystyle {\frac {1}{n}}p'(x)}$ is the characteristic polynomial of the following ${\displaystyle (n-1)\times (n-1)}$ submatrix of ${\displaystyle C}$:
${\displaystyle C_{n-1}={\begin{bmatrix}c_{0}&c_{n-1}&\dots &c_{3}&c_{2}\\c_{1}&c_{0}&c_{n-1}&&c_{3}\\\vdots &c_{1}&c_{0}&\ddots &\vdots \\c_{n-3}&&\ddots &\ddots &c_{n-1}\\c_{n-2}&c_{n-3}&\dots &c_{1}&c_{0}\\\end{bmatrix}}}$

(see [4] for the proof).

## Analytic interpretation

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

Consider vectors in ${\displaystyle \mathbf {R} ^{n}}$ as functions on the integers with period ${\displaystyle n}$, (i.e., as periodic bi-infinite sequences: ${\displaystyle \dots ,a_{0},a_{1},\dots ,a_{n-1},a_{0},a_{1},\dots }$) or equivalently, as functions on the cyclic group of order ${\displaystyle n}$ (${\displaystyle C_{n}}$ or ${\displaystyle \mathbf {Z} /n\mathbf {Z} }$) geometrically, on (the vertices of) the regular ${\displaystyle n}$-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 ${\displaystyle (c_{0},c_{1},\dots ,c_{n-1})}$; this is a discrete circular convolution. The formula for the convolution of the functions ${\displaystyle (b_{i}):=(c_{i})*(a_{i})}$ is

${\displaystyle b_{k}=\sum _{i=0}^{n-1}a_{i}c_{k-i}}$ (recall that the sequences are periodic)

which is the product of the vector ${\displaystyle (a_{i})}$ by the circulant matrix for ${\displaystyle (c_{i})}$.

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

The ${\displaystyle C^{*}}$-algebra of all circulant matrices with complex entries is isomorphic to the group ${\displaystyle C^{*}}$-algebra of ${\displaystyle \mathbf {Z} /n\mathbf {Z} }$.

## Symmetric circulant matrices

For a symmetric circulant matrix ${\displaystyle C}$ one has the extra condition that ${\displaystyle c_{n-i}=c_{i}}$. Thus it is determined by ${\displaystyle \lfloor n/2\rfloor +1}$ elements.

${\displaystyle C={\begin{bmatrix}c_{0}&c_{1}&\dots &c_{2}&c_{1}\\c_{1}&c_{0}&c_{1}&&c_{2}\\\vdots &c_{1}&c_{0}&\ddots &\vdots \\c_{2}&&\ddots &\ddots &c_{1}\\c_{1}&c_{2}&\dots &c_{1}&c_{0}\\\end{bmatrix}}.}$

The eigenvalues of any real symmetric matrix are real. The corresponding eigenvalues become:

${\displaystyle \lambda _{j}=c_{0}+2c_{1}\Re \omega _{j}+2c_{2}\Re \omega _{j}^{2}+\ldots +2c_{n/2-1}\Re \omega _{j}^{n/2-1}+c_{n/2}\omega _{j}^{n/2}}$

for ${\displaystyle n}$ even, and

${\displaystyle \lambda _{j}=c_{0}+2c_{1}\Re \omega _{j}+2c_{2}\Re \omega _{j}^{2}+\ldots +2c_{(n-1)/2}\Re \omega _{j}^{(n-1)/2}}$

for ${\displaystyle n}$ odd, where ${\displaystyle \Re z}$ denotes the real part of ${\displaystyle z}$. This can be further simplified by using the fact that ${\displaystyle \Re \omega _{j}^{k}=\cos(2\pi jk/n)}$.

## Complex symmetric circulant matrices

The complex version of the circulant matrix, ubiquitous in communications theory, is usually Hermitian. In this case ${\displaystyle c_{n-i}=c_{i}^{*},\;i\leq n/2}$ and its determinant and all eigenvalues are real.

If n is even the first two rows necessarily takes the form

${\displaystyle {\begin{bmatrix}r_{0}&z_{1}&z_{2}&r_{3}&z_{2}^{*}&z_{1}^{*}\\z_{1}^{*}&r_{0}&z_{1}&z_{2}&r_{3}&z_{2}^{*}\\\dots \\\end{bmatrix}}.}$

in which the first element ${\displaystyle r_{3}}$ in the top second half-row is real.

If n is odd we get

${\displaystyle {\begin{bmatrix}r_{0}&z_{1}&z_{2}&z_{2}^{*}&z_{1}^{*}\\z_{1}^{*}&r_{0}&z_{1}&z_{2}&z_{2}^{*}\\\dots \\\end{bmatrix}}.}$

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

## Applications

### In linear equations

Given a matrix equation

${\displaystyle \mathbf {C} \mathbf {x} =\mathbf {b} ,}$

where ${\displaystyle C}$ is a circulant square matrix of size ${\displaystyle n}$ we can write the equation as the circular convolution

${\displaystyle \mathbf {c} \star \mathbf {x} =\mathbf {b} ,}$

where ${\displaystyle c}$ is the first column of ${\displaystyle C}$, and the vectors ${\displaystyle c}$, ${\displaystyle x}$ and ${\displaystyle b}$ 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

${\displaystyle {\mathcal {F}}_{n}(\mathbf {c} \star \mathbf {x} )={\mathcal {F}}_{n}(\mathbf {c} ){\mathcal {F}}_{n}(\mathbf {x} )={\mathcal {F}}_{n}(\mathbf {b} )}$

so that

${\displaystyle \mathbf {x} ={\mathcal {F}}_{n}^{-1}\left[\left({\frac {({\mathcal {F}}_{n}(\mathbf {b} ))_{\nu }}{({\mathcal {F}}_{n}(\mathbf {c} ))_{\nu }}}\right)_{\!\nu \in \mathbf {Z} }\right]^{\rm {T}}.}$

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 (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

1. ^ Davis, Philip J., Circulant Matrices, Wiley, New York, 1970 ISBN 0471057711
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". New Zealand Journal of Mathematics. 36: 195–211.
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