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: