A stochastic matrix, also called a probability matrix, probability transition matrix, transition matrix, substitution matrix, or Markov matrix, is matrix used to characterize transitions for a finite Markov chain, 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 
stochastic matrices over 
(i.e., the integers mod 2),
![[1 0; 0 1] and [0 1; 1 0].](/images/equations/StochasticMatrix/NumberedEquation1.svg) |
There are six nonsingular stochastic 
matrices over 
,
![[0 1; 1 0],[0 2; 1 2],[1 0; 0 1],[1 2; 0 2],[2 0; 2 1],[2 1; 2 0].](/images/equations/StochasticMatrix/NumberedEquation2.svg) |
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 give the number of distinct stochastic matrices (and distinct nonsingular stochastic matrices) over 
for small 
.
 | stochastic  matrices over  |
| 2 | 1, 4, 64, 4096, ... |
| 3 | 1, 9, 729, ... |
| 4 | 1, 16, 4096, ... |
 | stochastic nonsingular  matrices over  |
| 2 | 1, 2, 24, 1440, ... |
| 3 | 1, 6, 450, ... |
| 4 | 1, 12, 3108, ... |
