Toeplitz matrix

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:

{\begin{bmatrix}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{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+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 only 2n−1 degrees of freedom, rather than n². 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 the Levinson algorithm in Θ(n²) time.^[1] Variants of this algorithm have been shown to be weakly stable (i.e. they exhibit numerical stability for well-conditioned linear systems).^[2] The algorithm can also be used to find the determinant of a Toeplitz matrix in O(n²) time.^[3]

A Toeplitz matrix can also be decomposed (i.e. factored) in O(n²) time.^[4] The Bareiss algorithm for an LU decomposition is stable.^[5] An LU decomposition gives a quick method for solving a Toeplitz system, and also for computing the determinant.

Algorithms that are asymptotically faster than those of Bareiss and Levinson have been described in the literature, but their accuracy cannot be relied upon.^[6]^[7]^[8]^[9]

General properties

A Toeplitz matrix may be defined as a matrix A where A_i,j = c_i−j, for constants c_1−n … c_n−1. The set of n×n Toeplitz matrices is a subspace of the vector space of n×n matrices under matrix addition and scalar multiplication.

Two Toeplitz matrices may be added in O(n) time and multiplied in O(n²) 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 commute asymptotically. This means they diagonalize in the same basis when the row and column dimension tends to infinity.

Nonsingular Toeplitz matrix A can be factored as:

A=VDU

where $D$ is a diagonal matrix, and $V$ and $U$ are two Vandermonde matrices: $(V)_{ij}=\gamma _{j}^{1-i}$ , $(U)_{ij}=\gamma _{i}^{j-1}$ . $\gamma _{1},\gamma _{2},...,\gamma _{n}$ are nonzero numbers.

For symmetric Toeplitz matrices, there are decomposition:

{\frac {1}{a_{0}}}A=GG^{T}-(G-I)(G-I)^{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}}}(BB^{T}-CC^{T})

where $B,C$ are two lower triangular Toeplitz matrices, and $C$ is strictly lower triangular. ^[10]

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={\begin{bmatrix}h_{1}&0&\ldots &0&0\\h_{2}&h_{1}&\ldots &\vdots &\vdots \\h_{3}&h_{2}&\ldots &0&0\\\vdots &h_{3}&\ldots &h_{1}&0\\h_{m-1}&\vdots &\ldots &h_{2}&h_{1}\\h_{m}&h_{m-1}&\vdots &\vdots &h_{2}\\0&h_{m}&\ldots &h_{m-2}&\vdots \\0&0&\ldots &h_{m-1}&h_{m-2}\\\vdots &\vdots &\vdots &h_{m}&h_{m-1}\\0&0&0&\ldots &h_{m}\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\\\vdots \\x_{n}\end{bmatrix}}

y^{T}={\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}&\vdots \\0&\ldots &0&0&0&x_{1}&\ldots &x_{n-2}&x_{n-1}&x_{n}\end{bmatrix}}.

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

Infinite Toeplitz matrix

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

A={\begin{bmatrix}\ldots &\ldots &\ldots &\ldots &\ldots &\ldots \\\ldots &a_{0}&a_{-1}&a_{-2}&a_{-3}&\ldots \\\ldots &a_{1}&a_{0}&a_{-1}&a_{-2}&\ldots \\\ldots &a_{2}&a_{1}&a_{0}&a_{-1}&\ldots \\\ldots &a_{3}&a_{2}&a_{1}&a_{0}&\ldots \\\ldots &\ldots &\ldots &\ldots &\ldots &\ldots \\\end{bmatrix}}.

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 $L^{\infty }$ norm of its symbol. The proof is easy to establish and can be found as Theorem 1.1 in the google book link: ^[11]

Notes

^ Press et al. 2007, §2.8.2—Toeplitz matrices
^ Krishna & Wang 1993
^ Monahan 2011, §4.5—Toeplitz systems
^ Brent 1999
^ Bojanczyk et al. 1995
^ Stewart 2003
^ Chen et al. 2006
^ Chan & Jin 2007
^ Chandrasekeran et al. 2007
^ Mukherjee, Bishwa Nath, and Sadhan Samar Maiti. “On Some Properties of Positive Definite Toeplitz Matrices and Their Possible Applications.” Linear Algebra and Its Applications 102 (1988): 211–240.
^ Böttcher & Grudsky 2012

References

Bojanczyk, A. W.; Brent, R. P.; de Hoog, F. R.; Sweet, D. R. (1995), "On the stability of the Bareiss and related Toeplitz factorization algorithms", SIAM Journal on Matrix Analysis and Applications, 16: 40–57, arXiv:1004.5510, doi:10.1137/S0895479891221563
Böttcher, Albrecht; Grudsky, Sergei M. (2012), Toeplitz Matrices, Asymptotic Linear Algebra, and Functional Analysis, Birkhäuser, ISBN 978-3-0348-8395-5
Brent, R. P. (1999), "Stability of fast algorithms for structured linear systems", in Kailath, T.; Sayed, A. H. (eds.), Fast Reliable Algorithms for Matrices with Structure, SIAM, pp. 103–116
Chan, R. H.-F.; Jin, X.-Q. (2007), An Introduction to Iterative Toeplitz Solvers, SIAM
Chandrasekeran, S.; Gu, M.; Sun, X.; Xia, J.; Zhu, J. (2007), "A superfast algorithm for Toeplitz systems of linear equations", SIAM Journal on Matrix Analysis and Applications, 29: 1247–1266, CiteSeerX 10.1.1.116.3297, doi:10.1137/040617200
Chen, W. W.; Hurvich, C. M.; Lu, Y. (2006), "On the correlation matrix of the discrete Fourier transform and the fast solution of large Toeplitz systems for long-memory time series", Journal of the American Statistical Association, 101: 812–822, doi:10.1198/016214505000001069
Krishna, H.; Wang, Y. (1993), "The Split Levinson Algorithm is weakly stable", SIAM Journal on Numerical Analysis, 30 (5): 1498–1508, doi:10.1137/0730078
Monahan, J. F. (2011), Numerical Methods of Statistics, Cambridge University Press
Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. (2007), Numerical Recipes: The Art of Scientific Computing (Third ed.), Cambridge University Press, ISBN 978-0-521-88068-8
Stewart, M. (2003), "A superfast Toeplitz solver with improved numerical stability", SIAM Journal on Matrix Analysis and Applications, 25: 669–693, doi:10.1137/S089547980241791X