Transition-rate matrix
In probability theory, a transition-rate matrix is an array of numbers describing the instantaneous rate at which a continuous-time Markov chain transitions between states.
In a transition-rate matrix, element denotes the rate departing from and arriving in state. The rates, and the diagonal elements are defined such that
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.
Properties
The transition-rate matrix has following properties:- There is at least one eigenvector with a vanishing eigenvalue, exactly one if the graph of is strongly connected.
- All other eigenvalues fulfill.
- All eigenvectors with a non-zero eigenvalue fulfill.
- The Transition-rate matrix satisfies the relation where P is the continuous stochastic matrix.