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:

Any n×n matrix A of the form

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

A Toeplitz matrix is not necessarily square.

Solving a Toeplitz systemEdit

A matrix equation of the form

 

is called a Toeplitz system if A is a Toeplitz matrix. If A is an   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 propertiesEdit

  • An n×n Toeplitz matrix may be defined as a matrix A where Ai,j = ci−j, for constants c1−ncn−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 (by storing only one value of each diagonal) 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.
  • A positive semi-definite n×n Toeplitz matrix   of rank r < n can be uniquely factored as
 
where   is an r×r positive definite diagonal matrix,   is an n×r Vandermonde matrix such that the columns are  . Here   and   is normalized frequency, and   is the Hermitian transpose of  . If the rank r = n, then the Vandermonde decomposition is not unique.[10]
  • For symmetric Toeplitz matrices, there is the decomposition
 
where   is the lower triangular part of  .
  • The inverse of a nonsingular symmetric Toeplitz matrix has the representation
 
where   and   are lower triangular Toeplitz matrices and   is a strictly lower triangular matrix.[11]

Discrete convolutionEdit

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   and   can be formulated as:

 
 

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

Infinite Toeplitz matrixEdit

A bi-infinite Toeplitz matrix (i.e. entries indexed by  )   induces a linear operator on  .

 

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

In such cases,   is called the symbol of the Toeplitz matrix  , and the spectral norm of the Toeplitz matrix   coincides with the   norm of its symbol. The proof is easy to establish and can be found as Theorem 1.1 in the google book link: [12]

See alsoEdit

  • Circulant matrix, a Toeplitz matrix with the additional property that  
  • Hankel matrix, an "upside down" (i.e., row-reversed) Toeplitz matrix

NotesEdit

ReferencesEdit

  • 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 (4): 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 (474): 812–822, CiteSeerX 10.1.1.574.4394, 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
  • 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
  • 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 (3): 669–693, doi:10.1137/S089547980241791X
  • Yang, Zai; Xie, Lihua; Stoica, Petre (2016), "Vandermonde decomposition of multilevel Toeplitz matrices with application to multidimensional super-resolution", IEEE Transactions on Information Theory, 62 (6): 3685–3701, arXiv:1505.02510, doi:10.1109/TIT.2016.2553041

Further readingEdit