Let $P$ be a transition matrix of a markov chain then we can compute the probabilities to get from state $i$ to state $j$ in $k$ moves by computing $(P^k)_{ij}$. If $P$ is diagonalizable then we can compute $B$, $D$ and $B^{-1}$ s.t. $P^k = BD^kB^{-1}$ which makes computation easier.
What if $P$ isn't diagonalizable? Is there any suitable way to compute the probabilities and is there any way to deduce the non-diagonalizability from the graph itself? I assume there is no stationary distribution at all.