transitiematrix

A transitiematrix, commonly referred to in English as a transition matrix, is a matrix used to describe the probabilities of moving between states in one step of a stochastic process, most often a discrete-time Markov chain. For a finite state space S = {1, ..., N}, the transition matrix P has entries p_ij = P(X_{t+1} = j | X_t = i). By construction, all entries are nonnegative and the rows sum to one, so P is a row-stochastic matrix. The matrix encodes how likely the process is to transition from each current state i to every possible next state j.

One-step transition probabilities generalize to n steps via P^n: (P^n)_{ij} = P(X_{t+n} = j | X_t = i). The study of P includes questions about long-run behavior, stationary distributions, and hitting times. A stationary distribution π is a row vector with πP = π and sum(π) = 1; it describes a distribution over states that remains unchanged under the transition dynamics. If the chain is irreducible and aperiodic, P^n converges to a matrix with identical rows equal to π, regardless of the initial state.

Special cases include absorbing transitions, where certain states are absorbing (p_ii = 1). For transient states, the fundamental matrix N = (I - Q)^{-1} (where Q is the submatrix of transient-to-transient transitions) gives expected times to absorption and related quantities.

In continuous-time models, the analogous object is the generator matrix Q; the transition probabilities over time t are given by P(t) = exp(Qt). The transitiematrix thus plays a central role in modeling, analysis, and computation of Markovian dynamics across disciplines.