siirtymämatriisi

A siirtymämatriisi, or transition matrix in English, is a fundamental concept in the study of Markov chains and other stochastic processes. It is a square matrix where each element represents the probability of transitioning from one state to another in a single step. The rows and columns of the matrix correspond to the possible states of the system.

The values within the transition matrix are always non-negative, and the sum of probabilities in each row must equal one, reflecting that from any given state, the system must transition to one of the possible states. If P is a transition matrix with states i and j, then P_ij represents the probability of moving from state i to state j.

The behavior of a Markov chain over multiple steps can be determined by raising the transition matrix to the power of the number of steps. For example, P^n gives the probabilities of transitioning between states after n steps. This allows for the prediction of long-term behavior, such as steady-state probabilities, where the distribution of states no longer changes over time.

Siirtymämatriisit are widely used in various fields, including physics, economics, biology, and computer science, to model systems that evolve probabilistically over discrete time steps. Their ability to capture the dynamics of state changes makes them a powerful tool for analysis and prediction.