Stochastic matrix

A **stochastic matrix** is a square matrix used in probability theory and statistics that describes a system where the probabilities of transitions from one state to another are represented. Each of its rows (or columns, depending on the type of stochastic matrix) sums to one, reflecting the fact that the total probability must equal one. There are two main types of stochastic matrices: 1. **Right Stochastic Matrix**: In a right stochastic matrix, each row sums to one.

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A row-stochastic matrix has nonnegative entries and each row sums to one. For a time-homogeneous Discrete-time Markov chain, its transition matrix has entries

P_{ij} = P (X_{n + 1} = j ∣ X_{n} = i) .

If a state has zero probability at time

n

, this row is specified by the model's Markov kernel, rather than recovered by conditioning on that zero-probability event. Rows index departure states and columns arrival states. Writing the current probability distribution as a row vector

μ_{n}

gives

μ_{n + 1} = μ_{n} P, μ_{n} = μ_{0} P^{n} .

The row-sum condition ensures that matrix multiplication preserves total probability. Under a column convention the evolution uses the transpose; the convention must accompany the formula.

P^{2} = (0.72 0.56 0.28 0.44) .

In particular,

(P^{2})_{SS} = 0.8 \times 0.8 + 0.2 \times 0.4 = 0.72

. The two terms sum the disjoint paths

S \to S \to S

and

S \to R \to S

. Requiring both subsequent days to be sunny has probability

0.64

; requiring only the second day to be sunny permits the extra path through rain.

Generally

(P^{n})_{ij}

is the

n

P^{m + n} = P^{m} P^{n}

is the discrete-time Chapman-Kolmogorov equation: sum over the intermediate state. This makes matrix powers useful for forecasting many steps without enumerating every path individually.