I have a problem(1) where the solution is more and more precise as I raise a matrix to a higher power. Both matrices and vector are over the real numbers, between 0.0 and 1.0.
I'm looking for $s$ in
$s = V \times M^x$ as $x \to \infty $
This converges, but I'd like to find the limit to which $s$ goes. $M$ sums to 1 on every row, if that helps. I denote vector-matrix multiplication by $\times$. Sorry if this is the wrong symbol. With V and M fixed, are there techniques I can use to compute this limit?
(1) V is a vector of probabilities of being in a given state, M is a transition matrix, giving the probability of going to each state from each state. The problem is to find the chance of being in specific states after infinite iterations.