Toeplitz matrix

In the mathematical discipline of 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:

$\begin{bmatrix} a & b & c & d & k \\ f & a & b & c & d \\ g & f & a & b & c \\ h & g & f & a & b \\ j & h & g & f & a \end{bmatrix}.$

Any n×n matrix A of the form

$A = \begin{bmatrix} a_{0} & a_{-1} & a_{-2} & \ldots & \ldots &a_{-n+1} \\ a_{1} & a_0 & a_{-1} & \ddots & & \vdots \\ a_{2} & a_{1} & \ddots & \ddots & \ddots& \vdots \\ \vdots & \ddots & \ddots & \ddots & a_{-1} & a_{-2}\\ \vdots & & \ddots & a_{1} & a_{0}& a_{-1} \\ a_{n-1} & \ldots & \ldots & a_{2} & a_{1} & a_{0} \end{bmatrix}$

is a Toeplitz matrix. If the i,j element of A is denoted A_i,j, then we have

A i, j = a i - j .

Properties

Generally, a matrix equation

A x = b

is the general problem of n linear simultaneous equations to solve. If A is a Toeplitz matrix, then the system is rather special (has only 2n − 1 degrees of freedom, rather than n²). One could therefore expect that solution of a Toeplitz system would be easier.

This can be investigated by the transformation

A U n - U m A,

which has rank 2, where $U k$ is the down-shift operator. Specifically, one can by simple calculation show that

$AU_n-U_mA= \begin{bmatrix} a_{-1} & \cdots & a_{-n+1} & 0 \\ \vdots & & & -a_{-n+1} \\ \vdots & & & \vdots \\ 0 & \cdots & & -a_{n-n-1} \end{bmatrix}$

where empty places in the matrix are replaced by zeros.

Notes

These matrices have uses in computer science because it can be shown that the addition of two Toeplitz matrices can be done in O(n) time, a Toeplitz matrix can be multiplied by a vector in O(n log n) time, and the matrix multiplication of two Toeplitz matrices can be done in O( $n 2$ ) time.

Toeplitz systems of form $A x = b$ can be solved by the Levinson-Durbin Algorithm in Θ( $n 2$ ) time. Variants of this algorithm have been shown to be weakly stable in the sense of James Bunch (i.e., they exhibit numerical stability for well-conditioned linear systems).

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.

If a Toeplitz matrix has the additional property that $a i = a i + n$ , then it is a circulant matrix.

Toeplitz matrices are persymmetric. Symmetric Toeplitz matrices are centrosymmetric.

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

Failed to parse (unknown function\ast): \begin{matrix}y & = & x \ast h \\ & = & \begin{bmatrix}h_1 & h_2 & h_3 & \ldots & h_{m-1} & h_m \\ \end{bmatrix} \begin{bmatrix}x_1 & x_2 & x_3 & \ldots & x_n & 0 & 0 & 0& \ldots & 0 \\ 0 & x_1 & x_2 & x_3 & \ldots & x_n & 0 & 0 & \ldots & 0 \\ 0 & 0 & x_1 & x_2 & x_3 & \ldots & x_n & 0 & \ldots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ldots & \vdots & \vdots & \ldots & 0 \\ 0 & \ldots & 0 & 0 & x_1 & \ldots & x_{n-2} & x_{n-1} & x_{n} & 0 \\ 0 & 0 & \ldots & 0 & 0 & x_1 & \ldots & x_{n-2} & x_{n-1} & x_{n} \\ \end{bmatrix} \\ & = & \begin{bmatrix}h_1 & 0 & 0 & \ldots & 0 & 0 \\ h_2 & h_1 & 0 & \ldots & 0 & 0 \\ h_3 & h_2 & h_1 & \ldots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ h_{m-1} & h_{m-2} & h_{m-3} & \ldots & h_1 & 0 \\ h_m & h_{m-1} & h_{m-2} & \ldots & h_2 & h_1 \\ \end{bmatrix} \begin{bmatrix}x_1 \\ x_2 \\ x_3 \\ \vdots \\ x_{n-1} \\ x_n \\ \end{bmatrix} \end{matrix}

Note that while the two matrix multiplications above result in the identical vector, the first is necessarily a row vector, while the second is a column vector.

This approach can be extended to compute autocorrelation, cross-correlation, moving average etc ^[1].

External link

^ Using Toeplitz matrices in MATLAB [1]

Toeplitz and Circulant Matrices: A Review, by R. M. Gray zh-yue:Toeplitz矩陣

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Toeplitz matrix

Properties

Notes

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External link

Toeplitz matrix

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