# 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^{}.

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 |