Stochastic matrix

In mathematics, computer science, and statistics, particularly in Linear algebra, Stochastic Matricies are square matricies whose columns are probability vectors, or add up to one.

Here is an example of a stochastic matrix P:

$P={\begin{bmatrix}.5&.2&.3\\.3&.8&.3\\.2&0&.4\end{bmatrix}}$

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

$Gh=h$

An example:

$G={\begin{bmatrix}.95&.03\\.05&.97\end{bmatrix}}$

and

$h={\begin{bmatrix}.375\\.625\end{bmatrix}}$

$Gh={\begin{bmatrix}.95&.03\\.05&.97\end{bmatrix}}{\begin{bmatrix}.375\\.625\end{bmatrix}}={\begin{bmatrix}.35625+.01875\\.01875+.60625\end{bmatrix}}={\begin{bmatrix}.375\\.625\end{bmatrix}}$

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

Take P from above as a stochastic matrix:

$P^{2}={\begin{bmatrix}.37&.26&.33\\.45&.70&.45\\.18&.04&.22\end{bmatrix}}$

The Stochastic Matrix Theorem says if A is an n by n matrix, then A has a steady-state vector t and 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