Matrix Equation Solution

Question

Does anybody know how to solve this matrix equation

\begin{equation} { I = S - RSQ, } \end{equation}

where $I$ is the identity matrix, $R$, $Q$ are known matrices, and $S$ is the unknown matrix.

Answer 1

Rewrite the equation as $$I=ISI-RSQ$$ and apply vectorization $$\eqalign{ {\rm vec}(I) = (I\otimes I-Q^T\otimes R)\,{\rm vec}(S) \cr {\rm vec}(S) = (I\otimes I-Q^T\otimes R)^{-1}\,{\rm vec}(I) \cr S = {\rm devec}\Big((I\otimes I-Q^T\otimes R)^{-1}\,{\rm vec}(I)\Big) \cr }$$ If you can transform the equation into a Sylvester equation, then you can use a more efficient solver like Bartels–Stewart instead of this brute force approach.

Answer 2

Suppose the following series converge, then $$S=I+RQ+R^2Q^2+\cdots$$