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Revision difference : stochastic matrix |
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Version 5 |
| \paragraph{Definition} |
\paragraph{Definition} |
| 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 \emph{stochastic} if $$\sum_{i\in I} p_{ij} = 1$$ |
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 \emph{stochastic} if $$\sum_{i\in I} p_{ij} = 1$$ |
| for every $j\in I$. We call $\mv{P}$ \emph{doubly stochastic} if, in addition, $$\sum_{j\in I} p_{ij} = 1$$ |
for every $j\in I$. We call $\mv{P}$ \emph{doubly stochastic} if, in addition, $$\sum_{j\in I} p_{ij} = 1$$ |
| for all $i\in I$. |
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. |
Equivalently, $\mv{P}$ is stochastic if every column is a distribution, and doubly stochastic if, in addition, every row is a distribution. |
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| Stochastic and doubly stochastic matrices are common in discussions of random processes, particularly Markov chains. |
Stochastic and doubly stochastic matrices are common in discussions of random processes, particularly Markov chains. |
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