rowstochastic

Row-stochastic, or a row-stochastic matrix, is a square matrix P with nonnegative entries in which the sum of the entries in every row equals 1. Each row of P represents a probability distribution over states, so the entry Pij gives the probability of moving from state i to state j in one step of a finite Markov chain or other stochastic process.

Key properties include that the vector of all ones is a right eigenvector of P with eigenvalue 1 (P times the all-ones vector equals the all-ones vector). The eigenvalues of a row-stochastic matrix lie within the unit disk, so their magnitudes do not exceed 1. If the matrix is irreducible and aperiodic, there exists a unique stationary distribution pi, a row vector with nonnegative entries summing to 1, satisfying pi P = pi. For any initial distribution x0, the sequence xk = x0 P^k converges to pi as k grows.

The stationary distribution can be interpreted as the long-run distribution of states under the Markov process governed by P. If the chain is reducible or periodic, convergence to a single stationary distribution may fail or result in convergence to a distribution supported on a subset of states.

Relation to column-stochastic matrices: a matrix is row-stochastic if its rows sum to 1, while its transpose is column-stochastic (columns sum to 1). In the row-stochastic setting, the stationary distribution is the left eigenvector pi of P corresponding to eigenvalue 1, i.e., pi P = pi; equivalently, it is the right eigenvector of P^T for eigenvalue 1.

Example: P = [[0.7, 0.3], [0.4, 0.6]] is row-stochastic. Its stationary distribution is pi = [4/7, 3/7], since [4/7, 3/7] P = [4/7, 3/7]. Row-stochastic matrices model transitions in finite Markov chains and other discrete-time stochastic processes.