Let $P$ be the transition matrix for a Markov chain. I understand what it means.
But what information can I get from matrix $P^2$, or $P^{\infty}$?
As I understand it now, $P^2$ holds the probabilities of getting from initial state $i$ (rows of matrix) and finishing in state $j$ (columns of matrix) in exactly 2 transitions.
So for $P^2 = \begin{pmatrix} 0.1 & 0.9 \\ 0.2 & 0.8 \\ \end{pmatrix}$, where states are $1,2$
There's 10% chance that if I begin in state 1, I will end up in state 1 in exactly two transitions; and 90% that I will end up in state 2.
Do I understand this correctly?
Thank you in advance.