Let $P$ be the transition matrix of a Markov chain, and assume that $P^2=P$. One immediate conclusion is that $P=P^\infty$.
Furthermore, assume that there is a state $i$ such as each state $j$ (including $j=i$), is accessible from $i$: $i\rightarrow j$.
I am fairly sure that this means that $P$ is irreducible (and thus, has only one stationary distribution, and all the columns of its matrix are equal), but I can't find a quick argument why this is true.
I think I can get a proof using the coefficients, to show that if there is a state $j$ such as $j\not\rightarrow i$, then $P^\infty\neq P$, because the probability of being in state $j$ will "grow over time", but I was wondering whether there was a more direct argument, perhaps making use of a standard result.