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.

Title	stochastic matrix
Canonical name	StochasticMatrix
Date of creation	2013-03-22 12:37:29
Last modified on	2013-03-22 12:37:29
Owner	mathwizard (128)
Last modified by	mathwizard (128)
Numerical id	9
Author	mathwizard (128)
Entry type	Definition
Classification	msc 60G99
Classification	msc 15A51
Related topic	Distribution
Related topic	Matrix
Defines	doubly stochastic
Defines	stochastic matrix