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 Mj 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.