Given numbers $\alpha \in \mathbb{R}$, $\beta, \omega \in \mathbb{C}$, we construct the matrix $H_n(\omega)$ that is defined as follows:
\begin{pmatrix} \alpha & \beta & \beta^2 & \ddots & \beta^{n-1} & \omega \\ \overline{\beta} & \alpha & \beta & \ddots & \ddots & \beta^{n-1} \\ \overline{\beta}^2 & \overline{\beta} & \alpha & \ddots & \beta^{2} & \ddots \\ \ddots & \ddots & \ddots & \ddots & \ddots & \beta^2 \\ \overline{\beta}^{n-1} & \ddots & \ddots & \ddots & \ddots & \beta \\ \overline{\omega} & \overline{\beta}^{n-1} & \ddots & \overline{\beta}^{2} & \overline{\beta} & \alpha \\ \end{pmatrix}
For which values of the parameter $\omega \in \mathbb{C}$ is the following hermitian matrix nonsingular? I also would like to determine whether the determinant is positive or negative and find the number of negative eigenvalues.