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Homestochastic matrix

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# 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.

Defines:

doubly stochastic, stochastic matrix

Related:

Distribution, Matrix

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

60G99*no label found*15A51

*no label found*

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## Info

## Attached Articles

## Corrections

meaning of random by armbrusterb ✓

nonnegativity by Koro ✓

Nonnegative. by Koro ✓

add classification by kshum ✓

nonnegativity by Koro ✓

Nonnegative. by Koro ✓

add classification by kshum ✓

## Versions

(v9) by mathwizard 2013-03-22