# Inverse of a Toeplitz Matrix

A Toeplitz matrix or diagonal-constant matrix is a matrix in which each descending diagonal from left to right is constant. For instance, the following matrix is an $n\times n$ Toeplitz matrix:

$$A = \begin{bmatrix} a_{0} & a_{-1} & a_{-2} & \ldots & \ldots &a_{-n+1} \\\ a_{1} & a_0 & a_{-1} & \ddots & & \vdots \\ a_{2} & a_{1} & \ddots & \ddots & \ddots& \vdots \\\ \vdots & \ddots & \ddots & \ddots & a_{-1} & a_{-2}\\\ \vdots & & \ddots & a_{1} & a_{0}& a_{-1} \\ a_{n-1} & \ldots & \ldots & a_{2} & a_{1} & a_{0} \end{bmatrix}$$

I'm interested in the self-adjoint case ($a_{-k}=a_{k}\in\mathbb{R}$).

My question are:

• Is there a relatively simple criterion to know when these matrices are invertible by just analyzing the sequence $\{a_{0},\ldots,a_{n-1}\}$?
• In the invertible case, what is known about its inverse?
If $\mathbf A$ is symmetric positive definite and Toeplitz, then there is an $O(n^2)$ method due to Trench for inverting it. So the SPD case is easy at least; what I'm fuzzy with is if (stable) methods for the symmetric indefinite case have been developed. (There are $O(n\log n)$ methods based on FFT, but I have no experience with using them.) –  Guess who it is. May 11 '11 at 12:53
@Tom: Interesting. Your comment also raised a good question. What are the conditions on the sequence $\{a_0,\ldots,a_{n-1}\}$ for $A$ to be positive definite? –  ght May 11 '11 at 14:19
You can apply the Gershgorin circle theorem to get a sufficient result for $A$ to be invertible, namely that it is strictly diagonally dominant $$|a_0| > \sum_{i=1}^{n-1} |a_i|$$ If this is true, a sufficient condition for it to be positive definite is that $a_0>0$. –  Calle May 11 '11 at 18:24