A square matrix with the elements on each row all nonnegative and summing to 1 is called *stochastic*.
The name comes from the "obvious" use in probability theory: the transition matrix of a Markov chain is a stochastic matrix (and every stochastic matrix is a transition matrix for a Markov chain).

As such, convergence theorems for Markov chains apply immediately to stochastic matrices. In particular, if no 1's appear in the matrix, then powers M^{j} of the matrix converge to a limit.

Stochastic matrices are also studied above fields other than **R**. If the field isn't an ordered field, the nonnegativity requirement will be dropped.

Doubly stochastic matrices are more interesting objects.