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

\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 (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