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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 eigenvector.

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 steady-state 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:

See also Muirhead's inequality.