In mathematics, a **stochastic matrix** (also termed **probability matrix**, **transition matrix**, **substitution matrix**, or **Markov matrix**) is a matrix used to describe the transitions of a Markov chain. Each of its entries is a nonnegative real number representing a probability. It has found use in probability theory, statistics and linear algebra, as well as computer science. There are several different definitions and types of stochastic matrices:

- A
**right stochastic matrix**is a square matrix of nonnegative real numbers, with each row summing to 1.

- A
**left stochastic matrix**is a square matrix of nonnegative real numbers, with each column summing to 1.

- A
**doubly stochastic matrix**is a square matrix of nonnegative real numbers with each row and column summing to 1.

In the same vein, one may define **stochastic vector** (also called **probability vector**) as a vector whose elements are nonnegative real numbers which sum to 1. Thus, each row of a right stochastic matrix (or column of a left stochastic matrix) is a stochastic vector.

A common convention in English language mathematics literature is to use row vectors of probabilities and right stochastic matrices rather than column vectors of probabilities and left stochastic matrices; this article follows that convention.

Read more about Stochastic Matrix: Definition and Properties, Example: The Cat and Mouse

### Other articles related to "stochastic matrix, matrix, stochastic":

**Stochastic Matrix**- Example: The Cat and Mouse - Phase-type Representation

... Let and remove state five to make a sub-

**stochastic matrix**, with where is the identity

**matrix**, and represents a column

**matrix**of all ones ...

... transition taking place, and with the initial state replaced by a

**stochastic**vector giving the probability of the automaton being in a given initial state ... The curried transition function can be understood to be a

**matrix**with

**matrix**entries The

**matrix**is then a square

**matrix**, whose entries are zero or one, indicating whether a transition is allowed by ... Such a transition

**matrix**is always defined for a non-deterministic finite automaton ...

... If the Markov chain is time-homogeneous, then the transition

**matrix**P is the same after each step, so the k-step transition probability can be computed as the k-th ... is 1) left eigenvector of the transition

**matrix**associated with the eigenvalue 1 ... transformation on the unit simplex associated to the

**matrix**P ...

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