For a matrix whose elements are stochastic, see Random matrix

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:

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.