Consider an irreducible and aperiodic finite state Markov chain with probability transition matrix $P$ with stationary distribution $\pi$. Denote by $\Pi$ the matrix $1\pi^T$, where $1$ is a column vector of all ones. Since $P\Pi = \Pi P = \Pi$, we have $P^t - \Pi = (P- \Pi)^t$.
Can someone explain how the last expression $P^t - \Pi = (P- \Pi)^t$ is derived?
Attempt: Say $t=2$, we would have $(P-\Pi)^2 = P^2 - 2 P \Pi + \Pi^2$. And repeated powers of stationary matrix $\Pi$ would give us $\Pi$ again, therefore $\Pi^2 = \Pi$. And the above expression reduces to $P^2 -\Pi$?
First show that $\Pi^2 = \Pi$. This is because $\pi^T 1 = 1$ because $\pi$ is a probability distribution.
Then proceed by recursion: trivial for $t=1$, then to go from $t$ to $t+1$ is easy using the equality i proved and the ones you mentioned.