There are several different definitions and types of stochastic matrices:

So that the total probability will be preserved, K is what is called a stochastic matrix.

Stochastic matrices are used to define Markov chains with finitely many states.

He also worked with Hazel Perfect on the spectra of doubly stochastic matrices.

A stochastic matrix describes a Markov chain over a finite state space S.

For some stochastic matrices "'P"', the limit does not exist, as shown by this example:

Since "'P"' is a row stochastic matrix, its largest left eigenvalue is 1.

Thus, a doubly stochastic matrix is both left stochastic and right stochastic.

A right stochastic matrix is a square matrix of nonnegative real numbers, with each row summing to 1.

Notice that the rows of P sum to 1: this is because P is a stochastic matrix.

A stochastic matrix is a matrix that has non-negative real entries that sum to one in each column, row, or both.

All 2-by-2 doubly stochastic matrices are unistochastic and orthostochastic, but for larger n it is not the case.

In probability theory, the probability m for the stochastic matrix obeys detailed balance when the stationary distribution has the property:

Furthermore, this can be generalized so that the entries in A can be real numbers representing connection strengths, as in a stochastic matrix.

A row of the stochastic matrix gives the probability distribution for the next position of some particle currently in the state that corresponds to the row.

A Google matrix is a particular stochastic matrix that is used by Google's PageRank algorithm.

The Perron-Frobenius theorem ensures that every stochastic matrix has such a vector, and that the largest absolute value of an eigenvalue is always 1.

The class of doubly stochastic matrices is a convex polytope known as the Birkhoff polytope .

A stochastic matrix is a square matrix of nonnegative real entries in which the sum of the entries in each column is 1.

The proof makes use of the fact that every doubly stochastic matrix is a weighted average of permutation matrices (Birkhoff-von Neumann theorem).

Regular stochastic matrix, a stochastic matrix such that all the entries of some power of the matrix are positive.

Alfred Horn, Doubly stochastic matrices and the diagonal of a rotation matrix, American Journal of Mathematics 76 (1954), 620-630.

In the mid 1960s, Mirsky's research focus shifted again, to combinatorics, after using Hall's marriage theorem in connection with his work on doubly stochastic matrices.

It is sometimes sufficient to use the matrix equation above and the fact that "'Q"' is a stochastic matrix to solve for "'Q"'.

A row (column) stochastic matrix is a square matrix each of whose rows (columns) consists of non-negative real numbers whose sum is unity.