Generator matrices
The generator matrix \(Q\) contains the instantaneous transition rates of a finite-state CTMC.
Entries
For \(i\ne j\), \(q_{ij}\ge0\) is the rate from state \(i\) to state \(j\). The diagonal entry is
\[q_{ii}=-\sum_{j\ne i}q_{ij},\]so each row of \(Q\) sums to zero.
Relation to transition probabilities
For a small time \(h\),
\[P(h)=I+hQ+o(h).\]Under standard finite-state assumptions,
\[P(t)=e^{tQ}.\]Key idea. A transition matrix contains probabilities; a generator matrix contains continuous-time transition rates.