Stochastic Matrix -- from Wolfram MathWorld

Stochastic Matrix

A stochastic matrix, also called a probability matrix, probability transition matrix, transition matrix, substitution matrix, or Markov matrix, is a square matrix used to characterize transitions for a finite Markov chain. Under the row convention, its entries are real numbers in the closed interval , and each row has sum 1.

In an algebraic usage, a 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 stochastic matrices over (i.e., the integers mod 2),

There are six nonsingular stochastic matrices over ,

In fact, the set of all nonsingular stochastic matrices over a field forms a group under matrix multiplication. This group is called the stochastic group.

The following tables use the same column-sum definition over the ring of integers modulo . They give the numbers of distinct stochastic matrices and distinct invertible matrices for , 2, ....