TITLE: Solving Linear Matrix Recurrences
QUESTION [5 upvotes]: Question:
Are there standard techniques available for solving  the following kind of linear matrix recurrence relations:
$$M_1,\cdots,M_k\ \in\ \mathbb{R}^{m\times n}$$
$$ A_1,\cdots,A_k\ \in\ \mathbb{R}^{n\times n}$$
$$M_{i+k+1}\ =\ \sum_{j=1}^{k}{M_{i+j}A_j} $$

I need to solve the special case of $k=2$ that appears in the iterative modification of edge weights of complete symmetric graphs.

REPLY [7 votes]: Same method as for scalar equations works. Put matrices $M_{n+k-1},M_{n+k-2},...,M_{n}$
vertically into a big matrix $X_n$ of size $mk\times n$. Then your recurrence becomes
a one-step recurrence $X_{n+1}=AX_{n}$, with some $A$, whose solution is $M^n=A^nX_0$, and this is solved by diagonalizing $A$ (or using its Jordan form).