Transitionsmatriser

Transitionsmatriser are square matrices used to describe the probabilities of moving between states in a stochastic process, most often in discrete-time Markov chains. Each entry P_ij represents the probability of transitioning from state i to state j in one time step. In the common row-stochastic convention, the sum of each row equals 1, reflecting total probability leaving state i. The matrix is nonnegative and often denoted by P or T.

P^n gives the n-step transition probabilities: the probability of being in state j after n steps when starting in state i is the (i,j) entry of P^n. A stationary distribution pi satisfies pi P = pi with sum pi_i = 1; for finite, irreducible, and aperiodic chains, P^n converges to a matrix with every row equal to pi, implying long-run stability.

Extensions include time-homogeneous versus time-inhomogeneous transition matrices; in the latter case, the matrix can vary with time. Absorbing states yield absorbing chains, where probabilities eventually concentrate on absorbing states. Transition matrices can be estimated from data by relative frequencies: P_ij ≈ N_ij / N_i, where N_ij counts observed i→j transitions and N_i is the total number of transitions out of i.

Applications span economics, genetics, queueing theory, computer science, and social sciences, where transitions between categories, states, or conditions are modeled, such as weather regimes, customer behavior, or web navigation. Related concepts include eigenvalues, the Perron–Frobenius eigenvector, and generator matrices for continuous-time processes.