Closed form solution to matrix geometric series

Question

Is there a closed form solution to the infinite sum $Q + AQA' + AAQA'A' + \ldots $ ?

Note: I'm assuming here all eigenvalues of the square matrix $A$ are less than one.

Answer 1

@echasnovski, that's perfect, thanks so much!

Just to recap:

Defining the sum of the first $k$ elements as $S_k$ and element $k$ as $M_k$ then

$S_1 + M_2 = A S_1 A^T + Q$

$S_2 + M_3 = A S_2 A^T + Q$

so that as $k \rightarrow \infty$

$S_\infty = A S_\infty A^T +Q$

The solution to this equation can then be found here