In probability theory, a transition-rate matrix (also known as a Q-matrix,^[1] intensity matrix,^[2] or infinitesimal generator matrix^[3]) is an array of numbers describing the instantaneous rate at which a continuous-time Markov chain transitions between states.

In a transition-rate matrix ${\displaystyle Q}$ (sometimes written ${\displaystyle A}$^[4]), element ${\displaystyle q_{ij}}$ (for ${\displaystyle i\neq j}$) denotes the rate departing from ${\displaystyle i}$ and arriving in state ${\displaystyle j}$. The rates ${\displaystyle q_{ij}\geq 0}$, and the diagonal elements ${\displaystyle q_{ii}}$ are defined such that

${\displaystyle q_{ii}=-\sum _{j\neq i}q_{ij}}$,

and therefore the rows of the matrix sum to zero.

Up to a global sign, a large class of examples of such matrices is provided by the Laplacian of a directed, weighted graph. The vertices of the graph correspond to the Markov chain's states.

The transition-rate matrix has following properties:^[5]

• There is at least one eigenvector with a vanishing eigenvalue, exactly one if the graph of ${\displaystyle Q}$ is strongly connected.
• All other eigenvalues ${\displaystyle \lambda }$ fulfill ${\displaystyle 0>\mathrm {Re} \{\lambda \}\geq 2\min _{i}q_{ii}}$.
• All eigenvectors ${\displaystyle v}$ with a non-zero eigenvalue fulfill ${\displaystyle \sum _{i}v_{i}=0}$.
• The Transition-rate matrix satisfies the relation ${\displaystyle Q=P'(0)}$ where P(t) is the continuous stochastic matrix.

An M/M/1 queue, a model which counts the number of jobs in a queueing system with arrivals at rate λ and services at rate μ, has transition-rate matrix

${\displaystyle Q={\begin{pmatrix}-\lambda &\lambda \\\mu &-(\mu +\lambda )&\lambda \\&\mu &-(\mu +\lambda )&\lambda \\&&\mu &-(\mu +\lambda )&\ddots &\\&&&\ddots &\ddots \end{pmatrix}}.}$

• Stochastic matrix

• Norris, J. R. (1997). Markov Chains. doi:10.1017/CBO9780511810633.005. ISBN 9780511810633.
• Suhov, Yuri; Kelbert, Mark (2008). Markov chains: a primer in random processes and their applications. Cambridge University Press.
• Syski, R. (1992). Passage Times for Markov Chains. IOS Press. ISBN 90-5199-060-X.

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