I am solving a matrix equation for the elements of the diagonal matrix $D$. Therefore, I must find vector real vector $e$, where $D = diag(e)$.
Vector $x$ and $y$ are given and are complex vectors. V is the matrix composed of the DFT vectors. Therefore, $VDV^{-1}$ is a circulant matrix representing a torus connectivity network. $F$ is diagonal and complex.
$$y = (F + VDV^{-1})^k \, x,$$
Is there an easy way to simplify the $k^{th}$ power of $\left(F + VDV^{-1}\right)$ so as to obtain an algebraic solution to this equation? Or would only numerical methods do the trick?
Thanks.