# Stochastic Matrix

A Stochastic Matrix is a square matrix that describes the transitions of a Markov Chain.

**AKA:**Probability/Transition Matrix.- …

**Counter-Example(s):****See:**Randomized Algorithm, Discrete-Time Discrete-Markov Process.

## References

### 2016

- (Wikipedia, 2016) ⇒ http://wikipedia.org/wiki/stochastic_matrix Retrieved:2016-3-22.
- 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, mathematical finance and linear algebra, as well as computer science and population genetics.There are several different definitions and types of stochastic matrices:

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**right stochastic matrix**is a real square matrix, with each row summing to 1.:A

**left stochastic matrix**is a real square matrix, 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.

- In mathematics, a