transitionsmatris

A transitionsmatris, or transition matrix, is a square matrix used in the study of stochastic processes to describe the probabilities of moving between states in one step. For a finite state space with n states, the entry P_ij is the probability of transitioning from state i to state j in one discrete time step. Depending on convention, rows sum to 1 (row-stochastic) and the state distribution evolves as p_{t+1} = p_t P, or columns may sum to 1 with P^T acting on column distributions.

Properties of a transitionsmatris include nonnegative entries and row sums equal to 1 in the common row-stochastic form. If the corresponding Markov chain is irreducible and aperiodic, there exists a unique stationary distribution pi satisfying pi = pi P and sum_i pi_i = 1. The chain’s long-run behavior can converge to this distribution from a wide range of initial states, provided the chain is ergodic. The matrix also reflects transient and recurrent classes, including absorbing states in special cases.

Construction and uses: A transitionsmatris can be built from data by counting observed transitions between states or specified by a model. It is fundamental in discrete-time Markov chains and has applications in finance, queueing theory, genetics, natural language processing, and various stochastic models. In practice, matrices are often manipulated to analyze steady states, mixing times, and other properties; variants include including damping factors, as in PageRank.

Computational notes: The stationary distribution is the eigenvector corresponding to eigenvalue 1, obtainable by the power method or other linear-algebra techniques. In continuous-time Markov chains, the analogous object is the transition rate matrix (generator) Q, with P(t) = exp(Qt); here rows sum to zero rather than one.