These questions are probably very basic but I'll dare to ask them anyway since I didn't have much luck in MathStackExchange.

Let $A$ be an $n\times n$ Hermitian Toeplitz matrix:

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

My questions are:

Is there a relatively "simple" criteria to determine if $A$ is invertible by analyzing the sequence $\{a_0, \ldots, a_{n-1} \}$?

Idem as before with positive definite?

In the invertible case, what is known about the structure of the inverse matrix? I seem to recall that this is well known.

What about the determinant?

Thanks!