Stochastic matrix

In mathematics, especially in probability theory and statistics, and also in linear algebra and computer science, a stochastic matrix is a square matrix whose columns are probability vectors which add up to one. It is the same thing as the matrix of transition probabilities of a finite Markov chain.

Here is an example of a stochastic matrix P:

If G is a stochastic matrix, then a steady-state vector or equilibrium vector for G is a probability vector h such that:

An example:

and

This case shows that Gh = 1h. For equations that show Gh = βh, for some real number β like Gh = 4h or Gh = -21h, see Eigenvectors.

A stochastic matrix is regular if some matrix power P^k contains only strictly positive entries.

Take P from above as a stochastic matrix:

Therefore, P is a regular stochastic matrix.

The Stochastic Matrix Theorem says if A is a regular stochastic matrix, then A has a steady-state vector t so that if x_o is any initial state and x_k+1 = Ax_k for k = 0,1,2,..... then the Markov chain {x_k} converges to t as k -> infinity. That is: