For a Markov chain $\{X_i\}$ with alphabet $\mathcal{X}$ of size $|\mathcal{X}|=m$ and transition matrix $Q=P_{ij}$, a distribution (or pmf) $\Pi$ on $\mathcal{X}$ is called stationary (or “steady state”) distribution for the MC if $\forall a\in\mathcal{X}$ $$\Pi(a)=\sum\limits_{b\in\mathcal{X}}\Pi(b)P_{ij}$$ or equivalently in matrix form $$\vec\Pi=\vec\Pi\cdot Q$$ $\Pi$ is row eigenvector of $Q$ with associated eigenvalue 1 ($\lambda\vec\Pi=\vec\Pi\cdot Q, \ \lambda=1$) where $$\vec\Pi:=(\Pi_{1},\cdots,\Pi_{m})$$where $$\Pi_{k}=\Pi(x_{k}), \ k=1,\cdots,m, \mbox{ and } \mathcal{X}=\{x_{1},\cdots,x_{m}\}$$