överföringsmatriser

Överföringsmatriser, also known as transition matrices or stochastic matrices, are fundamental tools in the study of discrete-time Markov chains. They are square matrices where each entry represents the probability of transitioning from one state to another in a single time step. The rows of an överföringsmatris sum to one, reflecting that from any given state, there is a certainty of moving to some state in the next step.

The elements of an överföringsmatris, denoted as P_ij, signify the probability of moving from state i to state j. For instance, if a system has three states (1, 2, 3), the överföringsmatris would be a 3x3 matrix. The entry in the first row and second column would be the probability of transitioning from state 1 to state 2.

Powers of an överföringsmatris provide information about transitions over multiple time steps. For example, the square of the matrix, P^2, gives the probabilities of moving between states in two time steps. As the number of steps increases, the rows of P^n tend to converge to the stationary distribution of the Markov chain, provided the chain is ergodic. This stationary distribution represents the long-term probabilities of being in each state. Överföringsmatriser are crucial for analyzing systems with probabilistic state changes, finding applications in fields such as physics, economics, biology, and computer science.