This is really great, but could you put in a bit how some transition matrices aren't markov (e.g. [0 1; 1 0]) and the convergence criterion where you can take M^n n->infinity and get the occupancy of the states?
This is an example of a Markov chain that is not aperiodic; what that means is that, given a starting node, at any point in time in the future, it will always be the case that it is impossible to be at a certain node. This ends up meaning that the Markov chain never ends up reaching a steady state; rather, its behavior is periodic!