Transzíciómátrixok

Transzíciómátrixok, or transition matrices, are square matrices used in probability theory and related fields to describe the transitions of a Markov chain between different states. Each entry in the matrix, denoted as P_ij, represents the probability of moving from state i to state j in a single time step. The rows of a transition matrix must sum to 1, reflecting the fact that from any given state, there is a 100% probability of transitioning to some state (including possibly staying in the same state).

The power of transition matrices lies in their ability to predict the long-term behavior of a Markov chain. If P is the transition matrix, then P^n gives the probabilities of transitioning between states in n time steps. As n approaches infinity, the matrix P^n often converges to a stationary distribution, where the probabilities of being in each state no longer change over time. This stationary distribution is a fundamental concept in analyzing the equilibrium behavior of systems modeled by Markov chains.

Applications of transition matrices are widespread, appearing in fields such as physics, chemistry, economics, biology, and computer science. They are used to model phenomena like radioactive decay, population dynamics, stock market fluctuations, gene evolution, and the random walks of particles. Understanding the properties and calculations involving transition matrices is crucial for analyzing and predicting the behavior of these complex systems. The dimensionality of the matrix corresponds to the number of possible states in the Markov chain.