Stochastic Matrix

A Stochastic matrix is the transition matrix for a finite Markov Chain, also called a Markov Matrix. Elements of the matrix must be Real Numbers in the Closed Interval [0, 1].

A completely independent type of stochastic matrix is defined as a Square Matrix with entries in a Field such that the sum of elements in each column equals 1. There are two nonsingular $2\times 2$ Stochastic Matrices over $\Bbb{Z}_2$ (i.e., the integers mod 2),

$\begin{displaymath} \left[{\matrix{1 & 0\cr 0 & 1\cr}}\right] \quad{\rm and}\quad \left[{\matrix{0 & 1\cr 1 & 0\cr}}\right]. \end{displaymath}$

There are six nonsingular stochastic $3\times 3$ Matrices over $\Bbb{Z}_3$ ,

$\begin{displaymath} \left[{\matrix{1 & 0\cr 0 & 1\cr}}\right], \left[{\matrix{0 ... ... & 2\cr}}\right], \left[{\matrix{1 & 2\cr 0 & 2\cr}}\right]\!. \end{displaymath}$

In fact, the set

of all nonsingular stochastic $n\times n$ matrices over a Field

forms a Group under Matrix Multiplication. This Group is called the Stochastic Group.

See also Markov Chain, Stochastic Group

References

Poole, D. G. ``The Stochastic Group.'' Amer. Math. Monthly 102, 798-801, 1995.