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),

There are six nonsingular stochastic matrices over ,