I an exercise, I have
$D=\text{diag}(8,\ldots,8) \in R^{10 \times 10}$, $M=\text{pentadiag}_{10}(-1,-1,1,0,0)$ and $N=M^t$.
I need to compute the spectral radius of the matrix $(M+D)^{-1}N$
Since $M+D$ is in Hessenberg form, the eigenvalues are easy to compute. I have that the only eigenvalue is $\lambda_1=9$ and therefore $(M+D)^{-1}$ has eigenvalue $\frac{1}{9}$. Also, $\lambda =1$ is the eigenvalue of $N$.
But with this I can't conclude. I think I should use some similarity trasformation, but I don't have any idea.
Any hint will be really appreciated