Circulant matrix

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.

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 $$C_n$$ 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 $$n \times n$$ circulant matrix $$C$$ takes the form $$C = \begin{bmatrix} c_0     & c_{n-1} & \cdots  & c_2     & c_1     \\ c_1     & c_0     & c_{n-1} &         & c_2     \\ \vdots  & c_1     & c_0     & \ddots  & \vdots  \\ c_{n-2} &         & \ddots  & \ddots  & c_{n-1} \\ c_{n-1} & c_{n-2} & \cdots  & c_1     & c_0     \\ \end{bmatrix}$$ or the transpose of this form (by choice of notation). If each $$c_i$$ is a $$p \times p$$ square matrix, then the $$np \times np$$ matrix $$C$$ is called a block-circulant matrix.

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

Different sources define the circulant matrix in different ways, for example as above, or with the vector $$c$$ 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 $$f(x) = c_0 + c_1 x + \dots + c_{n-1} x^{n-1}$$ is called the associated polynomial of the matrix $$C$$.

Eigenvectors and eigenvalues
The normalized eigenvectors of a circulant matrix are the Fourier modes, namely, $$v_j=\frac{1}{\sqrt{n}} \left(1, \omega^j, \omega^{2j}, \ldots, \omega^{(n-1)j}\right),\quad j = 0, 1, \ldots, n-1,$$ where $$\omega=\exp \left(\tfrac{2\pi i}{n}\right)$$ is a primitive $$n$$-th root of unity and $$i$$ 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 $$\lambda_j = c_0+c_{1} \omega^j + c_{2} \omega^{2j} + \dots + c_{n-1} \omega^{(n-1)j},\quad j = 0, 1, \dots, n-1.$$

Determinant
As a consequence of the explicit formula for the eigenvalues above, the determinant of a circulant matrix can be computed as: $$\det C = \prod_{j=0}^{n-1} (c_0 + c_{n-1} \omega^j + c_{n-2} \omega^{2j} + \dots + c_1\omega^{(n-1)j}).$$ Since taking the transpose does not change the eigenvalues of a matrix, an equivalent formulation is $$\det C = \prod_{j=0}^{n-1} (c_0 + c_1 \omega^j + c_2 \omega^{2j} + \dots + c_{n-1}\omega^{(n-1)j}) = \prod_{j=0}^{n-1} f(\omega^j).$$

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

Other properties
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}.$$ c_0    & c_{n-1} & \cdots  & c_3     & c_2     \\ c_1    & c_0     & c_{n-1} &         & c_3     \\ \vdots & c_1     & c_0     & \ddots  & \vdots  \\ c_{n-3} &        & \ddots  & \ddots  & c_{n-1} \\ c_{n-2} & c_{n-3} & \cdots & c_{1}   & c_0     \\ \end{bmatrix}$$ (see for the proof).
 * Any circulant is a matrix polynomial (namely, the associated polynomial) in the cyclic permutation matrix $$P$$: $$ C = c_0 I + c_1 P + c_2 P^2 + \dots + c_{n-1} P^{n-1} = f(P),$$ where $$P$$ is given by the companion matrix $$P = \begin{bmatrix}
 * The set of $$n \times n$$ circulant matrices forms an $$n$$-dimensional vector space with respect to addition and scalar multiplication. This space can be interpreted as the space of functions on the cyclic group of order $$n$$, $$C_n$$, or equivalently as the group ring of $$C_n$$.
 * Circulant matrices form a commutative algebra, since for any two given circulant matrices $$A$$ and $$B$$, the sum $$A + B$$ is circulant, the product $$AB$$ is circulant, and $$AB = BA$$.
 * For a nonsingular circulant matrix $$A$$, its inverse $$A^{-1}$$ is also circulant. For a singular circulant matrix, its Moore–Penrose pseudoinverse $$A^+$$ is circulant.
 * The matrix $$U$$ that is composed of the eigenvectors of a circulant matrix is related to the discrete Fourier transform and its inverse transform: $$ U_n^* = \frac{1}{\sqrt{n}} F_n, \quad\text{and}\quad U_n = \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.$$ Consequently the matrix $$U_n$$ diagonalizes $$C$$. In fact, we have $$ C = U_n \operatorname{diag}(F_n c) U_n^* = F_n^{-1} \operatorname{diag}(F_n c) F_n,$$ where $$c$$ is the first column of $$C$$. The eigenvalues of $$C$$ are given by the product $$F_n c$$. This product can be readily calculated by a fast Fourier transform. Conversely, for any diagonal matrix $$D$$, the product $$F_n^{-1}DF_n$$ is circulant.
 * Let $$p(x)$$ be the (monic) characteristic polynomial of an $$n \times n$$ circulant matrix $$C$$. Then the scaled derivative $\frac{1}{n}p'(x)$ is the characteristic polynomial of the following $$(n-1)\times(n-1)$$ submatrix of $$C$$: $$C_{n-1} = \begin{bmatrix}

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

Consider vectors in $$\R^n$$ as functions on the integers with period $$n$$, (i.e., as periodic bi-infinite sequences: $$\dots,a_0,a_1,\dots,a_{n-1},a_0,a_1,\dots$$) or equivalently, as functions on the cyclic group of order $$n$$ (denoted $$C_n$$ or $$\Z/n\Z$$) 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 $$(c_0,c_1,\dots,c_{n-1})$$; this is a discrete circular convolution. The formula for the convolution of the functions $$(b_i) := (c_i) * (a_i)$$ is
 * $$b_k = \sum_{i=0}^{n-1} a_i c_{k-i}$$

(recall that the sequences are periodic) which is the product of the vector $$(a_i)$$ by the circulant matrix for $$(c_i)$$.

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

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

Symmetric circulant matrices
For a symmetric circulant matrix $$C$$ one has the extra condition that $$c_{n-i}=c_i$$. Thus it is determined by $$\lfloor n/2\rfloor + 1$$ elements. $$C = \begin{bmatrix} c_0    & c_1 & \cdots & c_2    & c_1    \\ c_1    & c_0 & c_1    &        & c_2    \\ \vdots & c_1 & c_0    & \ddots & \vdots \\ c_2    &     & \ddots & \ddots & c_1    \\ c_1    & c_2 & \cdots & c_1    & c_0    \\ \end{bmatrix}.$$

The eigenvalues of any real symmetric matrix are real. The corresponding eigenvalues become: $$\lambda_j = c_0 + 2 c_1 \Re \omega_j + 2 c_2 \Re \omega_j^2 + \dots + 2c_{n/2-1} \Re \omega_j^{n/2-1} + c_{n/2} \omega_j^{n/2}$$ for $$n$$ even, and $$\lambda_j = c_0 + 2 c_1 \Re \omega_j + 2 c_2 \Re \omega_j^2 + \dots + 2c_{(n-1)/2} \Re \omega_j^{(n-1)/2}$$ for $$n$$ odd, where $$\Re z$$ denotes the real part of $$z$$. This can be further simplified by using the fact that $$\Re \omega_j^k = \cos(2\pi j k/n)$$ and $$\omega_j^{n/2}= \exp\left({\pi i j}\right)=\pm1$$ depending on $$j$$ 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 $$c_{n-i} = c_i^*, \; i \le n/2 $$ and its determinant and all eigenvalues are real.

If n is even the first two rows necessarily takes the form $$ \begin{bmatrix} r_0    & z_1 & z_2 & r_3 & z_2^* & z_1^* \\ z_1^* & r_0    & z_1 & z_2 & r_3 & z_2^*  \\ \dots \\ \end{bmatrix}. $$ in which the first element $$ r_3 $$ in the top second half-row is real.

If n is odd we get $$ \begin{bmatrix} r_0    & z_1 & z_2 & z_2^* & z_1^* \\ z_1^* & r_0   & z_1 & z_2 & z_2^* \\ \dots\\ \end{bmatrix}. $$

Tee has discussed constraints on the eigenvalues for the Hermitian condition.

In linear equations
Given a matrix equation
 * $$C \mathbf{x} = \mathbf{b},$$

where $$C$$ is a circulant matrix of size $$n$$, we can write the equation as the circular convolution $$\mathbf{c} \star \mathbf{x} = \mathbf{b},$$ where $$\mathbf c$$ is the first column of $$C$$, and the vectors $$\mathbf c$$, $$\mathbf x$$ and $$\mathbf b$$ 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 $$\mathcal{F}_{n}(\mathbf{c} \star \mathbf{x}) = \mathcal{F}_{n}(\mathbf{c}) \mathcal{F}_{n}(\mathbf{x}) = \mathcal{F}_{n}(\mathbf{b})$$ so that $$\mathbf{x} = \mathcal{F}_n^{-1} \left[ \left( \frac{(\mathcal{F}_n(\mathbf{b}))_{\nu}} {(\mathcal{F}_n(\mathbf{c}))_{\nu}} \right)_{\!\nu\in\Z}\, \right]^{\rm T}.$$

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/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.