I encountered a matrix power series: $$ X = M + PMP^{t} + P^{2}M(P^{t})^{2} + \cdots, $$ where $M$ is a real symmetric matrix, and $P$ is a real square matrix. Assuming that this series converges, multiplying the above by $P$ and $P^{t}$ from the left and right gives $$ PXP^{t} = PMP^{t} + P^{2}M(P^{t})^{2} + \cdots. $$ Hence it follows that $$ M = X - PXP^{t}, $$ but I couldn't find a way to solve this for $X$.
Of course, when $P$ and $M$ commute, we have $$ X = (1-PP^{t})^{-1}M. $$ Yet I don't want to make such an assumption.
Thanks in advance for help!