I have a arbitrary $N \times N$ matrix $S$. I have an function of this matrix given by:
$A = F(S) = 2S + P^{-1}S + 2SP + PSP$
where $P$ is the a cyclic permutation matrix which when acting on the left of a matrix sends row $i$ to row $i-1$ (with $0 = N$). For example with $N=3$
$P = \begin{pmatrix} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{pmatrix}.$
Thus $P^N = I$. I want to invert the above equation to find $ S = F^{-1}(A) $ which I am sure can be done.
Does anyone have any suggestions for how to do this most effectively?