stochastic matrix

Definition

Let $I$ be a finite or countable set, and let $𝐏=(p_{ij}:i,j\in I)$ be a matrix and let all $p_{ij}$ be nonnegative. We say $𝐏$ is stochastic if

$$\sum _{i\in I}{p}_{ij}=1$$

for every $j\in I$. We call $𝐏$ doubly stochastic if, in addition,

$$\sum _{j\in I}{p}_{ij}=1$$

for all $i\in I$.Equivalently, $𝐏$ is stochastic if every column is a distribution, and doubly stochastic if, in addition, every row is a distribution.

Stochastic and doubly stochastic matrices are common in discussions of random processes, particularly Markov chains.