transitiomatriisi

A transition matrix, also known as a stochastic matrix or Markov matrix, is a square matrix used to describe the probabilities of moving between states in a Markov chain. Each entry $P_{ij}$ in the matrix represents the probability of transitioning from state $i$ to state $j$. The rows of a transition matrix must sum to 1, indicating that from any given state, there is a certainty of transitioning to some state, including potentially remaining in the same state.

The columns of a transition matrix do not necessarily sum to 1. In a discrete-time Markov chain, the transition matrix is used to calculate the probability distribution of states at the next time step. If $v_t$ is a row vector representing the probability distribution of states at time $t$, then the distribution at time $t+1$, $v_{t+1}$, is given by $v_{t+1} = v_t P$, where $P$ is the transition matrix.

Transition matrices are fundamental in various fields, including probability theory, statistical mechanics, and economics. They are used to model systems that evolve over time through a series of probabilistic transitions, such as population dynamics, the spread of diseases, or the behavior of financial markets. The properties of a transition matrix, such as its eigenvalues and eigenvectors, can reveal long-term behavior of the system, including stationary distributions.