stochastic matrix

In mathematics, a stochastic matrix^[1] (also termed probability matrix,^[1] transition matrix,^[1][2] substitution matrix, or Markov matrix^[1]) is a square matrix^[1] used to describe the transitions of a Markov chain.^[1] Each of its entries is a nonnegative real number representing a probability.^[1] It 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] The stochastic matrix was first developed by Andrey Markov at the beginning of the 20th century.^[3] 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.^[1]

A left stochastic matrix is a real square matrix, with each column summing to 1.^[1]

A doubly stochastic matrix is a square matrix of nonnegative real numbers with each row and column summing to 1.^[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.^[1] Thus, each row of a right stochastic matrix (or column of a left stochastic matrix) is a stochastic vector.^[1]