# 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
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