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

Knowledge needed for this demonstration:

• Second property of the Matrix $H(t):

$$H(t)\cdot \vec{1}=\vec{1}$$

where $\vec{1}=[1,1,1,1,\dots ]$

• From the first part of the Demonstration on How to calculate the Matrix Q we know that:

$$\frac{d}{dt}H(t)=H(t)\cdot Q$$

So:

$$\begin{array}{rl}0& =\\ & =\frac{d}{dt}\vec{1}\\ & =\frac{d}{dt}\left(H(t)\cdot \vec{1}\right)\\ & =\frac{d}{dt}\left(H(t)\right)\cdot \vec{1}\\ & =Q\cdot H(t)\cdot \vec{1}\\ & =Q\cdot \vec{1}\end{array}$$

So we have that:

$$Q\cdot \vec{1}=0$$