Generalized Matrix Series I have to sum a matrix series of the form:
$$
\sum_{s=0}^\infty M^s B R^s
$$
Is it possible to obtain a closed form formula as in the usual geometric series?
Thanks.
 A: Let $A$ be the sum of the series, assuming it converges. Then 
$MAR = \sum_{s = 1}^\infty M^sBR^s = A - B$ and therefore
$$
MAR - A = -B \, .
$$
This matrix equation may be written as
$$
-R^T \otimes M \cdot vec(A) + vec(A) = vec(B)
$$
where $\otimes$ denotes the Kronecker product and $vec(A)$ is the vectorization of $A$ (stack all columns into a single columns, as in R). Therefore in this case (assuming invertibility)
$$
vec(A) = \left(I -  R^T \otimes M \right)^{-1} \cdot vec(B) 
$$
and you can get $A$ by unstacking the columns.
For this to work, the matrix $R^T \otimes M$ must have its spectrum inside the open unit disk. This means that $|\mu \cdot \mu'| < 1$ whenever $\mu$ is an eigenvalue of $M$ and $\mu'$ is an eigenvalue of $R$.  
A: Since $\|M^s B R^s\| \le \|M\|^s \|B\| \|R\|^s$ (using any sub-multiplicative matrix norm), the series converges if $\|M\| \|R\| < 1$
(this is a sufficient condition, not a necessary one).  The sum $S$ will have to satisfy 
$$ S = B + MSR $$
Since the function $S \mapsto B + MSR$ is a strict contraction, this uniquely defines $S$.  
Note that if $v$ is an eigenvector of $R$ for eigenvalue $\lambda$, 
$S v = B v + M S R v = B v + \lambda M S v$, so $S v = (I - \lambda M)^{-1} B v$.
