Toeplitz Matrix

Given 2n-1 numbers a_k, where k=-n+1, ..., -1, 0, 1, ..., n-1, a Toeplitz matrix is a matrix which has constant values along negative-sloping diagonals, i.e., a matrix of the form

 [a_0 a_(-1) a_(-2) ... a_(-n+1); a_1 a_0 a_(-1) ... |; a_2 a_1 a_0 ... a_(-2); | ... ... ... a_(-1); a_(n-1) ... a_2 a_1 a_0].

Matrix equations of the form


can be solved with O(n^2) operations. Typical problems modelled by Toeplitz matrices include the numerical solution of certain differential and integral equations (regularization of inverse problems), the computation of splines, time series analysis, signal and image processing, Markov chains, and queuing theory (Bini 1995).

See also

Triangular Matrix, Vandermonde Matrix

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Bini, D. "Toeplitz Matrices, Algorithms and Applications." ECRIM News Online Edition, No. 22, July 1995., W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Vandermonde Matrices and Toeplitz Matrices." §2.8 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 82-89, 1992.

Referenced on Wolfram|Alpha

Toeplitz Matrix

Cite this as:

Weisstein, Eric W. "Toeplitz Matrix." From MathWorld--A Wolfram Web Resource.

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