Given a positive constant $a\in\mathbb{R}$, , and a positive integer $n$, I am interested in the roots of $x^n + \sum_{i=0}^{n-1} a^i x^{n-i-1} = x^n + x^{n-1} + a x^{n-2} + a^2 x^{n-3} +\cdots + a^{n-1}= 0$. Images of the roots in the complex plane for a few select values of $n$ and $a$ are below. Is there analytical formula for these roots?
This problem is equivalent to finding the eigenvalues of the $n\times n$ Hessenberg matrix: $\left(\begin{matrix} -1& -1 & -1 & \cdots&-1 \\ a & 0 & 0& \cdots & 0 \\ 0 & a & 0 & \cdots &0 \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & \cdots & 0 & a &0 \end{matrix}\right)$
The geometric structure of the roots in the images makes me think there should be a clean analytic solution here, but I'm having difficulty to find one.
Note: Ultimately, I am trying to show that for $0<a<1$, the eigenvalues $\lambda$ of the following matrix satisfy $|\lambda|<1$. Numerically, I have found this to be true. Any ideas here would be appreciated as well.
$\left(\begin{matrix} -a& -1 & -1 & \cdots&-1 \\ a & 1-a & 0& \cdots & 0 \\ 0 & a & 1-a & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & a &1-a \end{matrix}\right)$