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 and linear algebra, as well as computer science. There are several different definitions and types of stochastic matrices:
- A right stochastic matrix is a square matrix of nonnegative real numbers, with each row summing to 1.
- A left stochastic matrix is a square matrix of nonnegative real numbers, 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.
Other articles related to "stochastic matrix, matrix, stochastic":
... Let and remove state five to make a sub-stochastic matrix, with where is the identity matrix, and represents a column matrix of all ones ...
... transition taking place, and with the initial state replaced by a stochastic vector giving the probability of the automaton being in a given initial state ... The curried transition function can be understood to be a matrix with matrix entries The matrix is then a square matrix, whose entries are zero or one, indicating whether a transition is allowed by ... Such a transition matrix is always defined for a non-deterministic finite automaton ...
... If the Markov chain is time-homogeneous, then the transition matrix P is the same after each step, so the k-step transition probability can be computed as the k-th ... is 1) left eigenvector of the transition matrix associated with the eigenvalue 1 ... transformation on the unit simplex associated to the matrix P ...
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