Toeplitz matrix#Solving a Toeplitz system

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:

: $\qquad\begin{bmatrix}
a & b & c & d & e \\
f & a & b & c & d \\
g & f & a & b & c \\
h & g & f & a & b \\
i & h & g & f & a
\end{bmatrix}.$

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

: $A = \begin{bmatrix}
a_0 & a_{-1} & a_{-2} & \cdots & \cdots & a_{-(n-1)} \\
a_1 & a_0 & a_{-1} & \ddots & & \vdots \\
a_2 & a_1 & \ddots & \ddots & \ddots & \vdots \\
\vdots & \ddots & \ddots & \ddots & a_{-1} & a_{-2} \\
\vdots & & \ddots & a_1 & a_0 & a_{-1} \\
a_{n-1} & \cdots & \cdots & a_2 & a_1 & a_0
\end{bmatrix}$

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 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.{{harvnb|Press| Teukolsky| Vetterling| Flannery| 2007 | loc= [http://apps.nrbook.com/empanel/index.html?pg=96 §2.8.2—Toeplitz matrices]}}{{harvnb|Hayes|1996|loc=Chapter 5.2.6}} Variants of the latter have been shown to be weakly stable (i.e. they exhibit numerical stability for well-conditioned linear systems).{{harvnb|Krishna | Wang |1993}} The algorithms can also be used to find the determinant of a Toeplitz matrix in $O(n^2)$ time.{{harvnb|Monahan |2011 | loc= §4.5—Toeplitz systems}}

A Toeplitz matrix can also be decomposed (i.e. factored) in $O(n^2)$ time.{{harvnb|Brent |1999}} The Bareiss algorithm for an LU decomposition is stable.{{harvnb|Bojanczyk|Brent|de Hoog|Sweet| 1995}} An LU decomposition gives a quick method for solving a Toeplitz system, and also for computing the determinant.

Properties

An $n\times n$ Toeplitz matrix may be defined as a matrix $A$ where $A_{i,j}=c_{i-j}$ , for constants $c_{1-n},\ldots,c_{n-1}$ . 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 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

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

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

The inverse of a nonsingular symmetric Toeplitz matrix has the representation

:: $A^{-1} = \frac{1}{\alpha_0} (B B^\operatorname{T} - C C^\operatorname{T})$

:where $B$ and $C$ are lower triangular Toeplitz matrices and $C$ is a strictly lower triangular matrix.{{harvnb|Mukherjee | Maiti |1988}}

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:

: $y = h \ast x =
\begin{bmatrix}
h_1 & 0 & \cdots & 0 & 0 \\
h_2 & h_1 & & \vdots & \vdots \\
h_3 & h_2 & \cdots & 0 & 0 \\
\vdots & h_3 & \cdots & h_1 & 0 \\
h_{m-1} & \vdots & \ddots & h_2 & h_1 \\
h_m & h_{m-1} & & \vdots & h_2 \\
0 & h_m & \ddots & h_{m-2} & \vdots \\
0 & 0 & \cdots & h_{m-1} & h_{m-2} \\
\vdots & \vdots & & h_m & h_{m-1} \\
0 & 0 & 0 & \cdots & h_m
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
\vdots \\
x_n
\end{bmatrix}$

: $y^T =
\begin{bmatrix}
h_1 & h_2 & h_3 & \cdots & h_{m-1} & h_m
\end{bmatrix}
\begin{bmatrix}
x_1 & x_2 & x_3 & \cdots & x_n & 0 & 0 & 0& \cdots & 0 \\
0 & x_1 & x_2 & x_3 & \cdots & x_n & 0 & 0 & \cdots & 0 \\
0 & 0 & x_1 & x_2 & x_3 & \ldots & x_n & 0 & \cdots & 0 \\
\vdots & & \vdots & \vdots & \vdots & & \vdots & \vdots & & \vdots \\
0 & \cdots & 0 & 0 & x_1 & \cdots & x_{n-2} & x_{n-1} & x_n & 0 \\
0 & \cdots & 0 & 0 & 0 & x_1 & \cdots & x_{n-2} & x_{n-1} & x_n
\end{bmatrix}.$

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

Infinite Toeplitz matrix

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

: $A=\begin{bmatrix}
& \vdots & \vdots & \vdots & \vdots \\
\cdots & a_0 & a_{-1} & a_{-2} & a_{-3} & \cdots \\
\cdots & a_1 & a_0 & a_{-1} & a_{-2} & \cdots \\
\cdots & a_2 & a_1 & a_0 & a_{-1} & \cdots \\
\cdots & a_3 & a_2 & a_1 & a_0 & \cdots \\
& \vdots & \vdots & \vdots & \vdots
\end{bmatrix}.$

The induced operator is bounded if and only if the coefficients of the Toeplitz matrix $A$ 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 can be found as Theorem 1.1 of Böttcher and Grudsky.{{harvnb|Böttcher|Grudsky|2012}}

Notes

References

{{citation | last1 = Bojanczyk | first1 = A. W. | last2 = Brent | first2 = R. P. | last3 = de Hoog | first3 = F. R. | last4 = Sweet | first4 = D. R. | year = 1995 | title = On the stability of the Bareiss and related Toeplitz factorization algorithms | journal = SIAM Journal on Matrix Analysis and Applications | volume = 16 | pages = 40–57 | doi = 10.1137/S0895479891221563| arxiv = 1004.5510 | s2cid = 367586 }}
{{citation|first1=Albrecht| last1= Böttcher| first2=Sergei M. | last2= Grudsky|title=Toeplitz Matrices, Asymptotic Linear Algebra, and Functional Analysis|url=https://books.google.com/books?id=Dmr0BwAAQBAJ&pg=PA1|year=2012|publisher=Birkhäuser|isbn=978-3-0348-8395-5}}
{{citation | first= R. P. | last= Brent | author-link=Richard P. Brent | year= 1999 | contribution= Stability of fast algorithms for structured linear systems| title= Fast Reliable Algorithms for Matrices with Structure | editor1-first= T. | editor1-last= Kailath | editor2-first= A. H. | editor2-last=Sayed | publisher= SIAM | pages=103–116|doi=10.1137/1.9781611971354.ch4| arxiv= 1005.0671 | hdl= 1885/40746 | isbn= 978-0-89871-431-9 | s2cid= 13905858 | hdl-access= free }}
{{citation|last1=Chan |first1=R. H.-F.| last2= Jin| first2= X.-Q. | year=2007 | title= An Introduction to Iterative Toeplitz Solvers | publisher= SIAM|doi=10.1137/1.9780898718850|isbn=978-0-89871-636-8 }}
{{citation | last1 = Chandrasekeran | first1 = S. | last2 = Gu | first2 = M. | last3 = Sun | first3 = X. | last4 = Xia | first4 = J. | last5 = Zhu | first5 = J. | year = 2007 | title = A superfast algorithm for Toeplitz systems of linear equations | journal = SIAM Journal on Matrix Analysis and Applications | volume = 29 | issue = 4| pages = 1247–66 | doi = 10.1137/040617200| citeseerx = 10.1.1.116.3297 }}
{{citation | last1 = Chen | first1 = W. W. | last2 = Hurvich | first2 = C. M. | last3 = Lu | first3 = Y. | year = 2006 | title = On the correlation matrix of the discrete Fourier transform and the fast solution of large Toeplitz systems for long-memory time series | journal = Journal of the American Statistical Association | volume = 101 | issue = 474| pages = 812–822 | doi = 10.1198/016214505000001069| citeseerx = 10.1.1.574.4394 | s2cid = 55893963 }}
{{citation | last=Hayes | first=Monson H.|title= Statistical digital signal processing and modeling | year = 1996 | publisher = Wiley | isbn = 0-471-59431-8}}
{{citation | last1 = Krishna | first1 = H. | last2=Wang | first2= Y. | title = The Split Levinson Algorithm is weakly stable | journal = SIAM Journal on Numerical Analysis | volume = 30 | issue = 5 | pages = 1498–1508 | year = 1993 | url = http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=SJNAAM000030000005001498000001&idtype=cvips&gifs=yes | doi = 10.1137/0730078| url-access = subscription }}
{{citation| last=Monahan | first= J. F. | year=2011 | title=Numerical Methods of Statistics | publisher= Cambridge University Press |isbn=978-1-139-08211-2 |doi=10.1017/CBO9780511977176}}
{{citation| last1=Mukherjee | first1= Bishwa Nath | first2= Sadhan Samar | last2= Maiti | year= 1988 | title=On some properties of positive definite Toeplitz matrices and their possible applications | journal= Linear Algebra and Its Applications | volume= 102 | pages= 211–240 | doi= 10.1016/0024-3795(88)90326-6 | url= https://core.ac.uk/download/pdf/82070573.pdf| doi-access= free }}
{{citation|last1=Press | first1=W. H. | last2= Teukolsky | first2= S. A. | last3= Vetterling | first3= W. T. | last4= Flannery | first4= B. P. | year=2007 | title= Numerical Recipes: The Art of Scientific Computing | edition= 3rd | publisher= Cambridge University Press | isbn= 978-0-521-88068-8| title-link=Numerical Recipes }}
{{citation | last = Stewart | first = M. | year = 2003 | title = A superfast Toeplitz solver with improved numerical stability | journal = SIAM Journal on Matrix Analysis and Applications | volume = 25 | issue = 3| pages = 669–693 | doi = 10.1137/S089547980241791X | s2cid = 15717371 }}
{{citation | last1 = Yang | first1 = Zai | last2 = Xie | first2 = Lihua | last3 = Stoica | first3 = Petre | year = 2016 | title = Vandermonde decomposition of multilevel Toeplitz matrices with application to multidimensional super-resolution | journal = IEEE Transactions on Information Theory | volume = 62 | issue = 6 | pages = 3685–3701 | doi = 10.1109/TIT.2016.2553041 | arxiv = 1505.02510 | s2cid = 6291005 }}