Let $\left(X_{r}\right)_{r \geqslant 0}$ be an irreducible, positive-recurrent Markov chain on the state space $S$ with transition matrix $\left(P_{i j}\right)$ and initial distribution $P\left(X_{0}=i\right)=\pi_{i}, i \in S$, where $\left(\pi_{i}\right)$ is the unique invariant distribution. What does it mean to say that the Markov chain is reversible?

Prove that the Markov chain is reversible if and only if $\pi_{i} P_{i j}=\pi_{j} P_{j i}$ for all $i, j \in S$.