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:
\begin{bmatrix}0.375 \\ 0.625 \end{bmatrix} = \begin{bmatrix} 0.35625 + 0.01875 \\ 0.01875 + 0.60625 \end{bmatrix} = \begin{bmatrix} 0.375 \\ 0.625 \end{bmatrix}</math>
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:
<math>\lim_{k \to \infty} A^k \textbf{x}_0 = \textbf{t}</math>
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