A square matrix with nonnegative entries whose rows sum to one, used to represent one-step transition probabilities of a discrete-time finite-state Markov chain under the row-convention.

Principle

Each row is a probability distribution: nonnegativity plus unit row-sums enforce conservation of total probability and permit composition of steps by matrix multiplication.

Demonstration

A 3×3 matrix P where P_{ij} is the probability of moving from state i to state j; the distribution after two steps is obtained by multiplying an initial row probability vector by P twice (vP^2).

Misapplication

Treating a matrix with negative entries or rows that do not sum to one as a stochastic matrix, or using a row-stochastic matrix directly as the infinitesimal generator of a continuous-time process without conversion.

Consequence

When applied correctly, repeated multiplication of a row probability vector by the matrix produces the discrete-time evolution of the chain; long-run behaviour is characterized by probability vectors π satisfying π = πP and by transition powers that describe multi-step probabilities.

Reversal

Interpreting the same nonnegative, unit-sum condition on columns yields a column-stochastic matrix, which represents a different convention (backward or column-wise transitions) and changes how state distributions are updated.

Boundary

Definition applies to finite, discrete-time, finite-dimensional matrices with nonnegative entries and unit row-sums; it excludes substochastic matrices, infinite-dimensional kernels without normalization, and continuous-time generators (infinitesimal rate matrices).

Semantic Tension

'Transition kernel' is a measure-theoretic generalization of the same idea for continuous or uncountable state spaces; a stochastic matrix is the finite, algebraic instance of such a kernel, not interchangeable when measure-theoretic structure matters.

Synthesis

A stochastic matrix is the finite-state, row-wise normalized linear operator that encodes one-step transition probabilities; its algebraic constraints (nonnegativity and row-sums equal to one) guarantee probabilistic composition and conservation when iterated.