Toeplitz matrix
1 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
$$\left[\begin{array}{ccccc}\hfill {a}_{0}\hfill & \hfill {a}_{1}\hfill & \hfill {a}_{2}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {a}_{n-1}\hfill \\ \hfill {a}_{-1}\hfill & \hfill {a}_{0}\hfill & \hfill {a}_{1}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill {a}_{-2}\hfill & \hfill {a}_{-1}\hfill & \hfill {a}_{0}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill {a}_{2}\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill {a}_{1}\hfill \\ \hfill {a}_{-(n-1)}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {a}_{-2}\hfill & \hfill {a}_{-1}\hfill & \hfill {a}_{0}\hfill \end{array}\right]$$ |
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 (http://planetmath.org/TimeComplexity).
References
- 1 Golub and Van Loan, Matrix Computations, Johns Hopkins University Press 1993
Title | Toeplitz matrix |
---|---|
Canonical name | ToeplitzMatrix |
Date of creation | 2013-03-22 13:04:22 |
Last modified on | 2013-03-22 13:04:22 |
Owner | akrowne (2) |
Last modified by | akrowne (2) |
Numerical id | 8 |
Author | akrowne (2) |
Entry type | Definition |
Classification | msc 15A09 |
Classification | msc 65F35 |
Classification | msc 15A57 |