Matrix used to describe the transitions of a Markov chain
For a matrix whose elements are stochastic, see 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.[1][2]: 9–11 It is also called a probability matrix, transition matrix, substitution matrix, or Markov matrix.[2]: 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.[2]: 1–8 There are several different definitions and types of stochastic matrices:[2]: 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.[2]: 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.[2]: 1–8 In addition, a substochastic matrix is a real square matrix whose row sums are all
^Asmussen, S. R. (2003). "Markov Chains". Applied Probability and Queues. Stochastic Modelling and Applied Probability. Vol. 51. pp. 3–8. doi:10.1007/0-387-21525-5_1. ISBN 978-0-387-00211-8.
^ abcdefGagniuc, Paul A. (2017). Markov Chains: From Theory to Implementation and Experimentation. USA, NJ: John Wiley & Sons. pp. 9–11. ISBN 978-1-119-38755-8.
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