I am looking for a formula for Partial Sum of a Geometric Matrix Series that has following structure: $S_n = A + M^TAM+(M^T)^2AM^2+...+ (M^T)^nAM^n$. I know that for a Series of $I + A + A^2 ...$ the formula is $S_n = (I-A)^{-1}(I-A^n)$. But I struggle to generalize it to the other case. Any suggestions would be very much appreciated!
Best regards and many thanks!
Mat