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Transition Matrix

A square matrix whose $(i,j)$ entry gives the probability of moving from state $i$ to state $j$ in one step of a Markov chain; each row sums to 1.

Stochastic ProcessesIntermediateFree

In Plain English

The transition matrix is a Markov chain's complete rulebook, written down as a grid of numbers. Row $i$ lists the probabilities of jumping from state $i$ to every other state in one step, and because the chain has to go somewhere, each row adds up to exactly 1. Multiply the matrix by itself and you get the two-step transition probabilities for free, this is why matrix algebra, not just probability reasoning, becomes the natural toolkit for studying Markov chains.

Definition

For a Markov chain with finite state space $S = {1, 2, \dots, k}$ , the transition matrix $P$ is the $k \times k$ matrix with entries $P_{ij} = p_{ij} = P (X_{n + 1} = j ∣ X_{n} = i)$ . Because every row represents a full probability distribution over the next state, $P$ is a (row-)stochastic matrix: all entries are non-negative and each row sums to 1, $\sum_{j \in S} p_{ij} = 1$ for every $i$ . The $n$ -step transition probabilities are given by matrix powers, $P^{(n)} = P^{n}$ , a direct consequence of the Chapman-Kolmogorov equation. If the chain has a stationary distribution $π$ (a row vector), it satisfies the eigenvector-like equation $π P = π$ .

Formula

P_{ij}^{(n)} = (P^{n})_{ij}, j \in S \sum P_{ij} = 1 for all i

Notation

P

—The one-step transition matrix, $k \times k$ for a $k$-state chain

P_{ij}

—Entry in row $i$, column $j$: the probability of moving from state $i$ to state $j$

π

—Stationary distribution, a row vector satisfying $\pi P = \pi$

Properties

Turning transition probabilities into a matrix is more than notation, it unlocks the entire machinery of linear algebra: eigenvalues determine the rate of convergence to the stationary distribution, and eigenvectors reveal the stationary distribution itself.
For a chain with a continuous or very large state space, the transition matrix generalizes to a transition kernel, a function or density rather than a finite grid of numbers, the underlying stochastic-matrix logic (rows integrating to 1) carries over unchanged.
A common student error is forgetting that rows, not columns, sum to 1 for a standard (row-stochastic) transition matrix, some texts use the transposed, column-stochastic convention instead, so always check which convention a given source uses before doing matrix arithmetic.

At a Glance

Shape

k \times k

square matrix for a

k

-state chain

Row constrainteach row sums to 1 (a valid probability distribution)

n-step probabilitiesgiven directly by the matrix power

P^{n}

Stationary distributionthe row vector

π

solving

π P = π

, if it exists

Last updated September 8, 2026← Back to the Dictionary