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?
- About its determinant?
Thanks!