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Toeplitz matrix
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(Definition)
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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
$$ \begin{bmatrix} a_0 & a_1 & a_2 & \cdots & a_{n-1} \\ a_{-1} & a_0 & a_1 & \ddots & \vdots \\ a_{-2} & a_{-1} & a_0 & \ddots & a_2 \\ \vdots & \ddots & \ddots & \ddots & a_1 \\ a_{-(n-1)} & \cdots & a_{-2} & a_{-1} & a_0 \end{bmatrix} $$
Numerical problems involving Toeplitz matrices typically have fast solutions (only $2n-1$ distinct elements need to be solved for, as opposed to $n^2$ . For example, the inverse of a symmetric, positive-definite $n\times n$ Toeplitz matrix can be found in $\mathcal{O}(n^2)$ time.
- 1
- Golub and Van Loan, Matrix Computations, Johns Hopkins University Press 1993
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"Toeplitz matrix" is owned by akrowne.
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Cross-references: symmetric, inverse, solutions, diagonal, matrix
There is 1 reference to this entry.
This is version 5 of Toeplitz matrix, born on 2002-09-28, modified 2005-08-14.
Object id is 3482, canonical name is ToeplitzMatrix.
Accessed 19461 times total.
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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Pending Errata and Addenda
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