Toeplitz matrix

Template:Short description In linear algebra, a Toeplitz matrix or diagonal-constant matrix, named after Otto Toeplitz, is a matrix in which each descending diagonal from left to right is constant. For instance, the following matrix is a Toeplitz matrix:

$\left[\begin{array}{ccccc}a& b& c& d& e\\ f& a& b& c& d\\ g& f& a& b& c\\ h& g& f& a& b\\ i& h& g& f& a\end{array}\right].$

Any $n\times n$ matrix $A$ of the form

$A=\left[\begin{array}{cccccc}{a}_0& a_{-1}& a_{-2}& \cdots & \cdots & a_{-(n-1)}\\ a_1& a_0& a_{-1}& \ddots & & ⋮\\ a_2& a_1& \ddots & \ddots & \ddots & ⋮\\ ⋮& \ddots & \ddots & \ddots & a_{-1}& a_{-2}\\ ⋮& & \ddots & a_1& a_0& a_{-1}\\ a_{n-1}& \cdots & \cdots & a_2& a_1& a_0\end{array}\right]$

is a Toeplitz matrix. If the $i,j$ element of $A$ is denoted $A_{i,j}$ then we have

$A_{i,j}=A_{i+1,j+1}=a_{i-j}.$

A Toeplitz matrix is not necessarily square.

Solving a Toeplitz system

A matrix equation of the form

$Ax=b$

is called a Toeplitz system if $A$ is a Toeplitz matrix. If $A$ is an $n\times n$ Toeplitz matrix, then the system has at most only $2n-1$ unique values, rather than $n^2$. We might therefore expect that the solution of a Toeplitz system would be easier, and indeed that is the case.

Toeplitz systems can be solved by algorithms such as the Schur algorithm or the Levinson algorithm in $O(n^2)$ time. Variants of the latter have been shown to be weakly stable (i.e. they exhibit numerical stability for well-conditioned linear systems). The algorithms can also be used to find the determinant of a Toeplitz matrix in $O(n^2)$ time.

A Toeplitz matrix can also be decomposed (i.e. factored) in $O(n^2)$ time. The Bareiss algorithm for an LU decomposition is stable. An LU decomposition gives a quick method for solving a Toeplitz system, and also for computing the determinant. Using displacement rank we obtain method requiring $\tilde{O}({\alpha }^{\omega -1}n)$ ops with the use of fast matrix multiplication algorithms, where $\alpha$ is the rank and ${}^{\sim }2.37\le \omega <3$.

Properties

• An $n\times n$ Toeplitz matrix may be defined as a matrix $A$ where $A_{i,j}=c_{i-j}$, for constants $c_{1-n},\dots ,c_{n-1}$. The set of $n\times n$ Toeplitz matrices is a subspace of the vector space of $n\times n$ matrices (under matrix addition and scalar multiplication).
• Two Toeplitz matrices may be added in $O(n)$ time (by storing only one value of each diagonal) and multiplied in $O(n^2)$ time.
• Toeplitz matrices are persymmetric. Symmetric Toeplitz matrices are both centrosymmetric and bisymmetric.
• Toeplitz matrices are also closely connected with Fourier series, because the multiplication operator by a trigonometric polynomial, compressed to a finite-dimensional space, can be represented by such a matrix. Similarly, one can represent linear convolution as multiplication by a Toeplitz matrix.
• Toeplitz matrices are asymptotically equivalent to circulant matrices as the dimension grows, a result known as the Grenander–Szegő theorem. This asymptotic circulant property is the reason that the discrete Fourier transform approximately diagonalizes large Toeplitz matrices, and underlies the effectiveness of DFT-based spectral density estimation for stationary processes.
• Toeplitz matrices commute asymptotically. This means they diagonalize in the same basis when the row and column dimension tends to infinity.
• For symmetric Toeplitz matrices, there is the decomposition

$\frac{1}{a_0}A=G{G}^{\mathrm{T}}-(G-I)(G-I{)}^{\mathrm{T}}$

where $G$ is the lower triangular part of $\frac{1}{a_0}A$.

• The inverse of a nonsingular symmetric Toeplitz matrix has the representation

$A^{-1}=\frac{1}{{\alpha }_0}(B{B}^{\mathrm{T}}-C{C}^{\mathrm{T}})$

where $B$ and $C$ are lower triangular Toeplitz matrices and $C$ is a strictly lower triangular matrix.

Discrete convolution

The convolution operation can be constructed as a matrix multiplication, where one of the inputs is converted into a Toeplitz matrix. For example, the convolution of $h$ and $x$ can be formulated as:

$y=h\ast x=\left[\begin{array}{ccccc}{h}_1& 0& \cdots & 0& 0\\ h_2& h_1& & ⋮& ⋮\\ h_3& h_2& \cdots & 0& 0\\ ⋮& h_3& \cdots & h_1& 0\\ h_{m-1}& ⋮& \ddots & h_2& h_1\\ h_m& h_{m-1}& & ⋮& h_2\\ 0& h_m& \ddots & h_{m-2}& ⋮\\ 0& 0& \cdots & h_{m-1}& h_{m-2}\\ ⋮& ⋮& & h_m& h_{m-1}\\ 0& 0& 0& \cdots & h_m\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\\ ⋮\\ x_n\end{array}\right]$

$y^T=\left[\begin{array}{cccccc}{h}_1& h_2& h_3& \cdots & h_{m-1}& h_m\end{array}\right]\left[\begin{array}{cccccccccc}{x}_1& x_2& x_3& \cdots & x_n& 0& 0& 0& \cdots & 0\\ 0& x_1& x_2& x_3& \cdots & x_n& 0& 0& \cdots & 0\\ 0& 0& x_1& x_2& x_3& \dots & x_n& 0& \cdots & 0\\ ⋮& & ⋮& ⋮& ⋮& & ⋮& ⋮& & ⋮\\ 0& \cdots & 0& 0& x_1& \cdots & x_{n-2}& x_{n-1}& x_n& 0\\ 0& \cdots & 0& 0& 0& x_1& \cdots & x_{n-2}& x_{n-1}& x_n\end{array}\right].$

This approach can be extended to compute autocorrelation, cross-correlation, moving average etc.

Infinite Toeplitz matrix

Template:Main A bi-infinite Toeplitz matrix (i.e. entries indexed by $\mathbb{Z}\times \mathbb{Z}$) $A$ induces a linear operator on ${\ell }^2$.

$A=\left[\begin{array}{ccccc}& ⋮& ⋮& ⋮& ⋮\\ \cdots & a_0& a_{-1}& a_{-2}& a_{-3}& \cdots \\ \cdots & a_1& a_0& a_{-1}& a_{-2}& \cdots \\ \cdots & a_2& a_1& a_0& a_{-1}& \cdots \\ \cdots & a_3& a_2& a_1& a_0& \cdots \\ & ⋮& ⋮& ⋮& ⋮\end{array}\right].$

The induced operator is bounded if and only if the coefficients of the Toeplitz matrix $A$ are the Fourier coefficients of some essentially bounded function $f$.

In such cases, $f$ is called the symbol of the Toeplitz matrix $A$, and the spectral norm of the Toeplitz matrix $A$ coincides with the $L^{\mathrm{\infty }}$ norm of its symbol. The proof can be found as Theorem 1.1 of Böttcher and Grudsky.

See also

• Circulant matrix, a square Toeplitz matrix with the additional property that $a_i=a_{i+n}$
• Hankel matrix, an "upside down" (i.e., row-reversed) Toeplitz matrix
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References

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Further reading

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