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.^: 9–11 It is also called a probability matrix, transition matrix, substitution matrix, or Markov matrix.^: 9–11 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.^: 1–8 There are several different definitions and types of stochastic matrices:^: 9–11

A 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 a 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.^: 9–11 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.^: 1–8 In addition, a substochastic matrix is a real square matrix whose row sums are all ${\displaystyle \leq 1.}$