If matrix $\sum_0^\infty C^k$ is convergent, how can I prove that $A(\sum_0^\infty C^k)B$ is convergent?

Question

For an $n \times n$ matrix $C$ and If $\sum_0^\infty C^k$ is convergent, how can I prove that for two matrices $A$ and $B$, $A(\sum_0^\infty C^k)B$ is convergent?

It seems quite obvious that you just multiply the convergent matrix sum with two constant matrices.

So, it has got to be convergent.

But I don't know how to prove it..

Answer 1

One approach is to note that the map $$ X \mapsto AXB $$ is continuous. As Marc van Leeuwen notes below, $$ A(\sum_{k=0}^N C^k)B = \sum_{k=0}^N AC^kB $$ so that the limit must be $\sum_{k=0}^\infty AC^kB$, which converges (by our statement of continuity).

Answer 2

If the series converges to $M = \sum_{k=0}^{\infty} C^k$ then one has, using as norm the sum of squares (or sum of absolute values if it is a complex matrix) of the elements of the matrix:

$$ ||A \sum_{k=0}^N C^k B - AMB|| = ||A(\sum_{k=0}^N C -M)B|| \leq ||A||||B|| \epsilon $$ for $N$ large enough, from whence the convergence follows.