Stochastic Matrix

A Stochastic Matrix is a square matrix that describes the transitions of a Markov Chain.

• AKA: Probability/Transition Matrix.
  • …
• Counter-Example(s):
  • a Random Matrix.
• See: Randomized Algorithm, Discrete-Time Discrete-Markov Process.

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References

2016

• (Wikipedia, 2016) ⇒ http://wikipedia.org/wiki/stochastic_matrix Retrieved:2016-3-22.
  • 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.

    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. Thus, each row of a right stochastic matrix (or column of a left stochastic matrix) is a stochastic vector.

    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.

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