DES - Demonstration of the Third Property of the Transition Rate Matrix Q

As we know from the Definition of ‘Transition Rate Matrix Q’

• The entries of $p_{i,j}$ are nonnegative for $j\ne i$
• And the sum of The sum along any row of $Q$ is $0$

As we stated in the demonstrations these properties are verified, but to be both true we have that the remaining entries ($p_{i,i}$) must be nonnegative

Also each entry $p_{i,i}$ have to be the sum of the $p_{i,j}$ for $i\ne j$ of the same row changed of sign, so:

$$p_{i,i}=-\sum _i{p}_{i,j}$$