Overgangsmatrix

Overgangsmatrix, or transition matrix in English, is a square matrix that describes the probabilities of moving from one state to another in a Markov chain. The element P_{ij} gives the probability of transitioning from state i to state j in one step. All entries are nonnegative and each row sums to one, making the matrix row-stochastic. In Danish usage, the term overgangsmatrix is used for this concept.

In a discrete-time Markov chain with finite or countable state space, the distribution after n steps is given by the initial distribution vector multiplied by P^n; the n-step transition probabilities are the entries of P^n.

If the chain is irreducible and aperiodic, powers of P converge to a matrix whose rows equal the stationary distribution pi, which satisfies pi P = pi and sum pi_i = 1. The stationary distribution represents a long-run proportion of time spent in each state.

The stationary distribution can be computed by solving the linear equations pi P = pi with sum pi_i = 1, or by finding the eigenvector of P associated with eigenvalue 1, typically by power iteration. For finite, well-behaved chains, these methods yield the same pi.

Continuous-time Markov chains use a generator matrix Q, and the transition matrix over time t is exp(Qt). The transition-matrix framework is widely used to model systems with probabilistic state changes, including queueing, economics, genetics, reliability, and stochastic processes in science and engineering.