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stochastic matrix
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(Definition)
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Let $I$ be a finite or countable set, and let $\mv{P} = (p_{ij} : i,j \in I)$ be a matrix and let all $p_{ij}$ be nonnegative. We say $\mv{P}$ is stochastic if $$\sum_{i\in I} p_{ij} = 1$$ for every $j\in I$ . We call $\mv{P}$ doubly stochastic if, in addition, $$\sum_{j\in I} p_{ij} = 1$$ for all $i\in I$ . Equivalently, $\mv{P}$ 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.
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"stochastic matrix" is owned by mathwizard. [ full author list (2) | owner history (1) ]
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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 13182 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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Pending Errata and Addenda
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