Toeplitz Matrix

A Toeplitz matrix is any $n\times n$ matrix with values constant along each (top-left to lower-right) diagonal. That is, a Toeplitz matrix has the form

$\displaystyle \begin{bmatrix} a_0 & a_1 & a_2 & \cdots & a_{n-1} \ a_{-1} & a... ...s & \ddots & a_1 \ a_{-(n-1)} & \cdots & a_{-2} & a_{-1} & a_0 \end{bmatrix}$

Numerical problems involving Toeplitz matrices typically have fast solutions (only distinct elements need to be solved for, as opposed to ). For example, the inverse of a symmetric, positive-definite $n\times n$ Toeplitz matrix can be found in $\mathcal{O}(n^2)$ time.

Bibliography

1: Golub and Van Loan, Matrix Computations, Johns Hopkins University Press 1993

"Toeplitz matrix" is owned by akrowne.

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Cross-references: symmetric, inverse, solutions, diagonal, matrix
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This is version 5 of Toeplitz matrix, born on 2002-09-28, modified 2005-08-14.
Object id is 3482, canonical name is ToeplitzMatrix.
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Classification:

AMS MSC:	65F35 (Numerical analysis :: Numerical linear algebra :: Matrix norms, conditioning, scaling)
	15A09 (Linear and multilinear algebra; matrix theory :: Matrix inversion, generalized inverses)
	15A57 (Linear and multilinear algebra; matrix theory :: Other types of matrices )

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Discussion

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block toeplitz matrices by silence on 2003-08-27 09:56:13

1. I am working with block Toeplitz matrices arising in the context of multivariate time-series data. I was wondering if anyone is aware of methods of qualitatively inspecting and analyzing such matrices? For example, what can one say about such matrices given their pseudospectra (spectral portrait)?

2. What exactly does the spectral radius (or even the full sequence of eigenvalues) convey about the matrix itself? That is, how does one interpret the concept of an "eigenvalue" for any matrix without talking about solving a linear system of equations and so on? Rephrased: Is there a qualitative or intuitive way to think about what eigenvalues represent in terms of the actual structure or geometry of a matrix?

3. In the case of block Toeplitz structures, suppose each block can be decomposed into at least one eigenvalue and associated eigenvector, then are the total number of eigenvalues in a block structure equivalent to the number of blocks necessarily?

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