PlanetMath: stochastic matrix

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stochastic matrix

(Definition)

Definition

Let

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

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

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

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

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.

"stochastic matrix" is owned by mathwizard. [ full author list (2) | owner history (1) ]

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Also defines:

doubly stochastic, stochastic matrix

Attachments:: there are no non-square doubly stochastic matrices (Result) by matte
eigenvalues of stochastic matrix (Theorem) by Andrea Ambrosio

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Cross-references: Markov chains, random processes, row, distribution, column, addition, matrix, countable, finite
There are 11 references to this entry.

This is version 6 of stochastic matrix, born on 2002-04-29, modified 2004-08-05.
Object id is 2885, canonical name is Stochastic.
Accessed 10903 times total.

Classification:

AMS MSC:	60G99 (Probability theory and stochastic processes :: Stochastic processes :: Miscellaneous)
	15A51 (Linear and multilinear algebra; matrix theory :: Stochastic matrices)

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