Given a diagonal matrix $\Lambda$ (with at most one entry equal to zero) and an arbitrary matrix $M$, is it possible to determine $S$ from the matrix equation $S = M + \Lambda S \Lambda$?
I do not have any experience in solving equations like this, so any suggestions on how to start would be most welcome. Recommendations for (introductory) literature on the topic would also be much appreciated.
This is very simple because $\Lambda$ is diagonal.
Write out $$ S = M + \Lambda S \Lambda $$ in coordinates $$ S_{ij} = M_{ij} + \sum_{kl} \Lambda_{ik} S_{kl} \Lambda_{lj} $$ Because $\Lambda$ is diagonal than this simplifies to $$ S_{ij} = M_{ij} + \Lambda_{ii} S_{ij} \Lambda_{jj} $$ Form this you get $$ S_{ij} = \frac{M_{ij}}{ 1- \Lambda_{ii}\Lambda_{jj}} $$ And you can see the condition under which your equation has a solution $$\Lambda_{ii}\Lambda_{jj} \neq 1 \qquad \text{or} \qquad M_{ij} = 0$$