Why a generator row sums to zero
Explain why the diagonal of a generator is non-positive and why each row sums to zero.
Worked answer (click to show)
Off-diagonal rates q_{ij} for j\neq i are non-negative: they are rates of jumps. The diagonal entry is defined to be minus their sum, q_{ii}=-\nu_i\le 0, which is the rate of leaving i. Each row then sums to \sum_{j\neq i}q_{ij}+q_{ii}=0. A row of probabilities would sum to 1. A row of rates sums to 0 because the outflow is recorded on the diagonal with a minus sign.