# stochastic matrix

## Definition

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

 $\sum_{i\in I}p_{ij}=1$

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

 $\sum_{j\in I}p_{ij}=1$

for all $i\in I$. Equivalently, $\mathbf{P}$ 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.

Title stochastic matrix StochasticMatrix 2013-03-22 12:37:29 2013-03-22 12:37:29 mathwizard (128) mathwizard (128) 9 mathwizard (128) Definition msc 60G99 msc 15A51 Distribution Matrix doubly stochastic stochastic matrix