Transition Matrix and Markov Chains

A transition matrix is a mathematical tool used to describe the transitions of a system from one state to another in various fields, such as probability, physics, and computer science. The concept is closely tied to Markov chains, which are models that represent random processes where the future state depends only on the current state and not on the sequence of events that preceded it.

Transition Matrix in Markov Chains

In the context of Markov chains, a transition matrix, often referred to as a stochastic matrix, is a square matrix used to describe the probabilities of moving from one state to another in a stochastic process. Each entry in the matrix represents the probability of moving from one state to another, with the sum of each row equaling one. This characteristic ensures that the matrix is row-stochastic.

A Markov chain is defined by a state space, a transition matrix, and an initial state. The transition matrix governs the dynamics of the chain, providing a compact representation of the possible transitions. For example, in a simple weather model, states might include "sunny," "cloudy," and "rainy," with the transition matrix indicating the likelihood of moving from one weather condition to another.

Properties of Transition Matrices

Stochastic Properties: A transition matrix is considered stochastic if all its entries are non-negative and each row sums up to one. This property ensures that it fulfills the criteria of representing probabilities.
Types of Matrices: There are various types of stochastic matrices, including doubly stochastic matrices, where both rows and columns sum to one, and regular stochastic matrices, which eventually reach a steady state where the probabilities stabilize.
Absorbing States: In some Markov chains, certain states are known as absorbing states, meaning once reached, the process remains in those states. The transition matrix for such chains features rows with a single '1' at the diagonal position and zeros elsewhere.

Applications

Economics: Transition matrices are used in modeling economic systems and market behavior, allowing economists to predict the movement between different economic states.
Biology: In population genetics, transition matrices are employed to predict changes in allele frequencies over generations.
Computer Science: Algorithms such as PageRank, used by search engines like Google, rely on transition matrices to rank web pages by importance.

Transition-Rate Matrices

A related concept is the transition-rate matrix, also known as a Q-matrix or intensity matrix, used in continuous-time Markov chains. Unlike discrete-time processes, where transitions occur at fixed intervals, continuous-time Markov chains allow transitions to occur at any time. The transition-rate matrix describes the rates at which transitions occur rather than their probabilities.

Connections to Other Mathematical Concepts