Choose the row-stochastic convention: $P_{ij}$ is the probability of moving from state $i$ to state $j$, so $P\mathbf1=\mathbf1$. For a row probability vector $\pi_k$, $\pi_{k+1}=\pi_kP$. For a column probability vector $q_k=\pi_k^T$, $q_{k+1}=P^Tq_k$.
A stationary column distribution satisfies $P^Tq=q$, $q\ge0$, $\mathbf1^Tq=1$. For $P=\begin{bmatrix}0.9&0.1\\0.5&0.5\end{bmatrix}$, $0.1q_1=0.5q_2$ gives $q=(5/6,1/6)^T$.
A finite irreducible chain has a unique stationary distribution; aperiodicity additionally ensures convergence to it from every initial distribution. For a regular matrix (some power strictly positive), both properties hold. A stationary vector exists for any finite stochastic matrix, but uniqueness and convergence need these extra hypotheses.
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