$$ \begin{Bmatrix} a & b & 0 &\dots & 0 & c \\ c & 0 & b &\dots & 0 & 0 \\ 0 & c & \ddots &\ddots & 0 & 0 \\ \vdots & 0 & \ddots &\ddots & b & 0 \\ \vdots & 0 & \ddots &\ddots & 0 & b \\ b & 0 & \dots &0 & c & 0 \\ \end{Bmatrix} $$
where $b, c \in \Bbb R$ and $b\ne c$. I would like to solve the eigenvalues of this $n \times n$ matrix, and see the impact of $a$ on its eigenvalues. This circulant matrix models a physical problem with BCs, finding an analytical solution allows to understand the asymptotic cases, for instance,
$a=0$, then the eigenvalues are supposed to be complex (and are known)
$a \to \inf$ and purely imaginary, the eigenvalues should be real.
I tried already some methods, for instance to do it iteratively, but unfortunately this wasn't helpful.