Circulant 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 ${\displaystyle C_{n}}$ and hence frequently appear in formal descriptions of spatially invariant linear operations. This property is also critical 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 ${\displaystyle n\times n}$ circulant matrix ${\displaystyle C}$ takes the form

or the transpose of this form (by choice of notation). When the term ${\displaystyle c_{i}}$ is a ${\displaystyle p\times p}$ square matrix, then the ${\displaystyle np\times np}$ matrix ${\displaystyle C}$ is called a block-circulant matrix.

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,

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 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 ${\displaystyle C}$ 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 ${\displaystyle P}$:
  $${\displaystyle C=c_{0}I+c_{1}P+c_{2}P^{2}+\dots +c_{n-1}P^{n-1}=f(P),}$$

  
  where ${\displaystyle P}$ is given by
  $${\displaystyle P={\begin{bmatrix}0&0&\cdots &0&1\\1&0&\cdots &0&0\\0&\ddots &\ddots &\vdots &\vdots \\\vdots &\ddots &\ddots &0&0\\0&\cdots &0&1&0\end{bmatrix}}.}$$

  
• 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:
  $${\displaystyle U_{n}^{*}={\frac {1}{\sqrt {n}}}F_{n},\quad {\text{and}}\quad U_{n}={\frac {1}{\sqrt {n}}}F_{n}^{-1},{\text{ where }}F_{n}=(f_{jk}){\text{ with }}f_{jk}=e^{-2jk\pi i/n},\,{\text{for }}0\leq j,k<n.}$$

  
  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 ${\textstyle {\frac {1}{n}}p'(x)}$ is the characteristic polynomial of the following ${\displaystyle (n-1)\times (n-1)}$ submatrix of ${\displaystyle C}$:
  $${\displaystyle C_{n-1}={\begin{bmatrix}c_{0}&c_{n-1}&\cdots &c_{3}&c_{2}\\c_{1}&c_{0}&c_{n-1}&&c_{3}\\\vdots &c_{1}&c_{0}&\ddots &\vdots \\c_{n-3}&&\ddots &\ddots &c_{n-1}\\c_{n-2}&c_{n-3}&\cdots &c_{1}&c_{0}\\\end{bmatrix}}}$$

  
  (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 ${\displaystyle \mathbb {R} ^{n}}$ 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 ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$) 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

(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 ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$.

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.

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

for ${\displaystyle n}$ even, and 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 ${\displaystyle \Re \omega _{j}^{k}=\cos(2\pi jk/n)}$.

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 ${\displaystyle c_{n-i}=c_{i}^{*},\;i\leq n/2}$ and its determinant and all eigenvalues are real.

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

in which the first element ${\displaystyle r_{3}}$ 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[]

In linear equations[]

Given a matrix equation

where ${\displaystyle C}$ is a circulant square matrix of size ${\displaystyle n}$ we can write the equation as the circular convolution where ${\displaystyle \mathbf {c} }$ is the first column of ${\displaystyle C}$, and the vectors ${\displaystyle \mathbf {c} }$, ${\displaystyle \mathbf {x} }$ and ${\displaystyle \mathbf {b} }$ 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 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 (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[]

External links[]

• R. M. Gray, Toeplitz and Circulant Matrices: A Review doi:10.1561/0100000006
• Weisstein, Eric W. "Circulant Matrix". MathWorld.
• IPython Notebook demonstrating properties of circulant matrices