Solution of a equation of matrices

Question

Can I find an explicit solution for the $R$ satisfying the equation $$\sum \limits_{k=1}^{\infty}R^kS(R^k)^T = M,$$ where $S$ and $M$ are known real, symmetric, and square matrices.

Any help will be greatly appreciated.

Answer 1

Define a new matrix variable $$A=R\otimes R$$ Vectorize each side of the equation to obtain $$\eqalign{ {\rm vec}(M) &= \sum_{k=1}^\infty A^k {\,\rm vec}(S) \cr m &= \sum_{k=1}^\infty A^k\,s = \Big(\frac{A}{I-A}\Big)\,s \cr (I-A)\,m &= As \cr m &= A(s+m) \cr }$$ De-vectorizing the last equation yields $$\eqalign{ M &= R(S+M)R^T \cr }$$ Now, can you solve that equation?