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 steadystate 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 steadystate 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: