AHL 4.19 (HL)—Transition matrices and Markov chains

A transition matrix stores probabilities of moving between states. With a column-vector convention, v_(n+1)=Pv_n; each column must sum to one. Powers of P give multi-step transition probabilities.

The orientation is part of the model: row-vector conventions transpose the multiplication order. A steady state is a distribution π with Pπ=π and entries summing to one, if the chain converges to one.

If 70% of users remain in state A and 30% move to B each month, the first column of P records those two destinations under the chosen convention. Multiply by the current state vector before interpreting the next month.

Matrix entries are not percentages to add across unrelated rows, and a stationary distribution is not guaranteed for every chain. State the convention, initial vector and convergence assumption.

A transition diagram labels the same probabilities encoded in $T$. A regular Markov chain has some power of $T$ with all positive entries and approaches a unique steady state independent of the initial state. Find long-run probabilities by repeated multiplication or solve $T\pi=\pi$ with entries summing to 1; $\pi$ is the eigenvector for eigenvalue 1 after normalization.