Toeplitz matrix

A matrix equation of the form

${\displaystyle Ax=b}$

is called a Toeplitz system if ${\displaystyle A}$ is a Toeplitz matrix. If ${\displaystyle A}$ is an ${\displaystyle n\times n}$ Toeplitz matrix, then the system has at-most only ${\displaystyle 2n-1}$ unique values, rather than ${\displaystyle n^{2}}$. 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 the Levinson algorithm in ${\displaystyle O(n^{2})}$ 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 ${\displaystyle O(n^{2})}$ time.[3]

A Toeplitz matrix can also be decomposed (i.e. factored) in ${\displaystyle O(n^{2})}$ 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]

• An ${\displaystyle n\times n}$ Toeplitz matrix may be defined as a matrix ${\displaystyle A}$ where ${\displaystyle A_{i,j}=c_{i-j}}$, for constants ${\displaystyle c_{1-n},\ldots ,c_{n-1}}$. The set of ${\displaystyle n\times n}$ Toeplitz matrices is a subspace of the vector space of ${\displaystyle n\times n}$ matrices (under matrix addition and scalar multiplication).
• Two Toeplitz matrices may be added in ${\displaystyle O(n)}$ time (by storing only one value of each diagonal) and multiplied in ${\displaystyle O(n^{2})}$ 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 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

${\displaystyle {\frac {1}{a_{0}}}A=GG^{\operatorname {T} }-(G-I)(G-I)^{\operatorname {T} }}$

where ${\displaystyle G}$ is the lower triangular part of ${\displaystyle {\frac {1}{a_{0}}}A}$.

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

${\displaystyle A^{-1}={\frac {1}{\alpha _{0}}}(BB^{\operatorname {T} }-CC^{\operatorname {T} })}$

where ${\displaystyle B}$ and ${\displaystyle C}$ are lower triangular Toeplitz matrices and ${\displaystyle C}$ is a strictly lower triangular matrix.[10]

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 ${\displaystyle h}$ and ${\displaystyle x}$ can be formulated as:

${\displaystyle y=h\ast x={\begin{bmatrix}h_{1}&0&\cdots &0&0\\h_{2}&h_{1}&&\vdots &\vdots \\h_{3}&h_{2}&\cdots &0&0\\\vdots &h_{3}&\cdots &h_{1}&0\\h_{m-1}&\vdots &\ddots &h_{2}&h_{1}\\h_{m}&h_{m-1}&&\vdots &h_{2}\\0&h_{m}&\ddots &h_{m-2}&\vdots \\0&0&\cdots &h_{m-1}&h_{m-2}\\\vdots &\vdots &&h_{m}&h_{m-1}\\0&0&0&\cdots &h_{m}\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\\\vdots \\x_{n}\end{bmatrix}}}$

${\displaystyle y^{T}={\begin{bmatrix}h_{1}&h_{2}&h_{3}&\cdots &h_{m-1}&h_{m}\end{bmatrix}}{\begin{bmatrix}x_{1}&x_{2}&x_{3}&\cdots &x_{n}&0&0&0&\cdots &0\\0&x_{1}&x_{2}&x_{3}&\cdots &x_{n}&0&0&\cdots &0\\0&0&x_{1}&x_{2}&x_{3}&\ldots &x_{n}&0&\cdots &0\\\vdots &&\vdots &\vdots &\vdots &&\vdots &\vdots &&\vdots \\0&\cdots &0&0&x_{1}&\cdots &x_{n-2}&x_{n-1}&x_{n}&0\\0&\cdots &0&0&0&x_{1}&\cdots &x_{n-2}&x_{n-1}&x_{n}\end{bmatrix}}.}$

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

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

${\displaystyle A={\begin{bmatrix}&\vdots &\vdots &\vdots &\vdots \\\cdots &a_{0}&a_{-1}&a_{-2}&a_{-3}&\cdots \\\cdots &a_{1}&a_{0}&a_{-1}&a_{-2}&\cdots \\\cdots &a_{2}&a_{1}&a_{0}&a_{-1}&\cdots \\\cdots &a_{3}&a_{2}&a_{1}&a_{0}&\cdots \\&\vdots &\vdots &\vdots &\vdots \end{bmatrix}}.}$

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

In such cases, ${\displaystyle f}$ is called the symbol of the Toeplitz matrix ${\displaystyle A}$, and the spectral norm of the Toeplitz matrix ${\displaystyle A}$ coincides with the ${\displaystyle L^{\infty }}$ norm of its symbol. The proof is easy to establish and can be found as Theorem 1.1 of: [11]

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

• 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, S2CID 367586
• 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, doi:10.1137/1.9781611971354.ch4, hdl:1885/40746, S2CID 13905858
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• Monahan, J. F. (2011), Numerical Methods of Statistics, Cambridge University Press
• Mukherjee, Bishwa Nath; Maiti, Sadhan Samar (1988), "On some properties of positive definite Toeplitz matrices and their possible applications" (PDF), Linear Algebra and Its Applications, 102: 211–240, doi:10.1016/0024-3795(88)90326-6
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• Bareiss, E. H. (1969), "Numerical solution of linear equations with Toeplitz and vector Toeplitz matrices", Numerische Mathematik, 13 (5): 404–424, doi:10.1007/BF02163269, S2CID 121761517
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