Transition-rate matrix

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.