Finite column-stochastic matrices and probability vectors

Let $S=\{1,\dots ,n\}$ with $n\ge 1$. A matrix $P=(P_{ij}{)}_{i,j\in S}$ is column-stochastic when $P_{ij}\ge 0$ and ${\sum }_{i\in S}{P}_{ij}=1$ for every source index $j$. A column probability vector is $x\in {\mathbb{R}}^n$ with $x_i\ge 0$ and ${\sum }_i{x}_i=1$. A stationary probability vector for $P$ is a column probability vector $\pi$ satisfying $P\pi =\pi$. For $z\in {\mathbb{R}}^n$, write $\mathrm{\parallel }z{\mathrm{\parallel }}_1={\sum }_i\mathrm{\mid }{z}_i\mathrm{\mid }$ and $H=\{z:{\sum }_i{z}_i=0\}$.

The published transition-matrix convention Transition matrices and n-step probabilities is row-stochastic: its row index is the source and its column index the destination. If $K$ is such a matrix, then $P=K^{\mathsf{T}}$ is column-stochastic; a row law $u$ updated by $uK$ corresponds to the column law $u^{\mathsf{T}}$ updated by $P{u}^{\mathsf{T}}$. All displays on this page that use $Px$ or $Qx$ use this column convention.