Stochastic matrix
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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, i.e., the entries in each column are nonnegative real numbers whose sum is 1. 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:
- <math>P = \begin{bmatrix}
0.5 & 0.2 & 0.3 \\ 0.3 & 0.8 & 0.3 \\ 0.2 & 0 & 0.4 \end{bmatrix}<math>
If G is a stochastic matrix, then a steady-state vector or equilibrium vector for G is a probability vector h such that:
- <math> Gh = h <math>
An example:
- <math>G = \begin{bmatrix}
0.95 & 0.03 \\ 0.05 & 0.97 \end{bmatrix}<math> and
- <math>h = \begin{bmatrix}
0.375 \\ 0.625 \end{bmatrix}<math>
- <math>Gh = \begin{bmatrix}
0.95 & 0.03 \\ 0.05 & 0.97 \end{bmatrix}
\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 eigenvector.
A stochastic matrix P is regular if some matrix power Pk contains strictly positive entries.
Using stochastic matrix P, from above:
- <math>P^2 = \begin{bmatrix}
0.37 & 0.26 & 0.33 \\ 0.45 & 0.70 & 0.45 \\ 0.18 & 0.04 & 0.22 \end{bmatrix}<math>
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 xo is any initial state and xk+1 = Axk for k = 0, 1, 2, ..... then the Markov chain {xk} converges to t as k -> infinity. That is:
<math>\lim_{k \to \infty} A^k \textbf{x}_0 = \textbf{t}.<math>
See also Muirhead's inequality.