Editing Stochastic matrix (section)

{{Short description|Matrix used to describe the transitions of a Markov chain}}
{{Use dmy dates|date=August 2021}}
{{For|a matrix whose elements are stochastic|Random matrix}}
In mathematics, a '''stochastic matrix''' is a [[square matrix]] used to describe the transitions of a [[Markov chain]]. Each of its entries is a [[nonnegative]] [[real number]] representing a [[probability]].<ref>{{Cite book | doi = 10.1007/0-387-21525-5_1 | first1 = S. R. | last1 = Asmussen| chapter = Markov Chains | title = Applied Probability and Queues | series = Stochastic Modelling and Applied Probability | volume = 51 | pages = 3–8 | year = 2003 | isbn = 978-0-387-00211-8 }}</ref><ref name=":1">{{Cite book |last=Lawler |first=Gregory F. |author-link=Greg Lawler |title=Introduction to Stochastic Processes |publisher=CRC Press |year=2006 |isbn=1-58488-651-X |edition=2nd |language=en}}</ref>{{Rp|page=10}} It is also called a '''probability matrix''', '''transition matrix''', ''[[substitution matrix]]'', or '''Markov matrix'''. The stochastic matrix was first developed by [[Andrey Markov]] at the beginning of the 20th century, and has found use throughout a wide variety of scientific fields, including [[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:

*A '''right stochastic matrix''' is a square matrix of nonnegative real numbers, with each row summing to 1 (so it is also called a '''row stochastic matrix''').
*A '''left stochastic matrix''' is a square matrix of nonnegative real numbers, with each column summing to 1 (so it is also called a '''column stochastic matrix''').
*A ''[[doubly stochastic matrix]]'' is a square matrix of nonnegative real numbers with each row and column summing to 1.
*A '''substochastic matrix''' is a real square matrix whose row sums are all <math>\le1.</math>
In the same vein, one may define a [[probability vector]] as a [[Euclidean vector|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 probability vector.  Right stochastic matrices act upon [[row vector]]s of probabilities by multiplication from the right (hence their name) and the matrix entry in the {{mvar|i}}-th row and {{mvar|j}}-th column is the probability of transition from state {{mvar|i}} to state {{mvar|j}}.  Left stochastic matrices act upon [[column vector]]s of probabilities by multiplication from the left (hence their name) and the matrix entry in the {{mvar|i}}-th row and {{mvar|j}}-th column is the probability of transition from state {{mvar|j}} to state {{mvar|i}}. 

This article uses the right/row stochastic matrix convention.