$E$ is eigenvalues of $M_n$ if and only if $W_n=|M_n-E I_n|=0$ where $I_n$ is a $n-by-n$ identity matrix. I have a closed form of such equation as
$$W_n=(-1)^{n/2}(b_1-b_2^{n-2}V_1V_2)$$
when $n=2m$. Also, $b_1, b_2, V_1$and $V_2$ is real parameter. The initial condition is $W_1=V_2$.
How can I obtain the eigenvalues of such a matrix?