According to wiki this is the definition: def of transition rate matrix
That I do not get is why each row needs to sum up to 0, and why $q_{i,i}$ is negative $\lambda_i.$
Thank you
2026-02-24 03:30:41.1771903841
Intuition on transition rate matrix of Continuous Time Markov Chain
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Imagine you have a amount of matter on a vertex $V_i$.
The line $q_{i,j}$ indicate the variation of the matter which were on $V_i$ is going. As nothing is create and nothing disappear, the mass of matter is still the same after the operation. So the line sum up to $0$.
And because the vertex $V_i$ can only loose matter, $q_{i,i}$ is negative.