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 questions 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!

About the last point of you question I think it's not really simple to state a closed simple formula for the determinants. I tried to see if there are symmetries. There are but not really useful (at least as far as I can see).

Just to have an idea the first 3 steps of the induction you have the following determinants.

If $n=2$ then

\begin{equation} \det A^{(2)} = a_0^2 - a_1^2 \end{equation}

If $n=3$ then

\begin{equation} \det A^{(3)} = (a_0 - a_2) (a_0^2 - 2 a_1^2 + a_0 a_2) \end{equation}

Already at $n=4$ the formula is not so simple. Indeed if we have $n=4$ we the determinant become:

\begin{equation} \det A^{(4)} = a_0^4 + a_1^4 + a_2^4 - 2 a_1 a_2^2 a_3 +4 a_0 a_1^2 a_2 - 3 a_1^2a_0^2 -2 a_2^2a_1^2 - 2 a_2^2a_0^2 + a_3^2a_1^2 - a_3^2a_0^2 - 2 a_1^3 a_3 + 4 a_0 a_1 a_2 a_3 \end{equation}

What happens is that when you have to calculate $\det A^{(n)}$ the minor determinants of order $n-1$ are matrices where the condition of symmetry that allowed huge simplifications disappears. With n=2,3 the problem is not so big since the minors are trivial, but when n gets bigger the problems arise. In fact the minors are not really Toeplitz matrix, but "block Toeplitz Matriz" (sort of saying). So maybe there could be a way of enclosing the writing in a simple notation formula, but it wouldn't be a real computational gain...