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 Pk 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 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:
Referenced By
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