DES - Definition of 'Transition Rate Matrix Q'

Transition Rate Matrix Q:

Let:

$$Q\triangleq \underset{dt\to 0}{\lim }\frac{H(dt)-I}{dt}$$

Matrix $Q$ is called transition rate matrix, because its entries have the dimension of $\left[{\text{time}}^{-1}\right]$

The entry of $Q$ in position $(i,j)$ will be denoted $q_{i,j}$

$$Q\triangleq {\left[q_{i,j}\right]}_{i,j\in \mathcal{X}}=\left[\begin{array}{cccc}{q}_{1,1}& q_{1,2}& q_{1,3}& \dots \\ q_{2,1}& q_{2,2}& q_{2,3}& \dots \\ q_{2,1}& q_{3,2}& q_{3,3}& \dots \\ ⋮& ⋮& ⋮& \ddots \end{array}\right]$$

Properties of $Q$

1. The entries $q_{i,j}$ of $Q$ are nonnegative for $j\ne i$. Demonstration.
2. The sum along any row of $Q$ is $0$, it can be also said that: $Q\cdot \vec{1}=0$ Demonstration.
3. The entries $q_{i,i}$ of $Q$ are nonpositive for $j\ne i$. Demonstration.

Relation with the matrix $H(t)$

Definition of the Transition Function Matrix: $H(t)$ 1.

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

$$H(t)=e^{Qt}$$

$$p_{i,j}=-\frac{q_{i,j}}{q_{i,i}}$$

Relation with the mean State Holding Time $E\left[V(i)\right]$

DES - Definition of ‘State Holding Time’ of the State Holding Time: $V(i)$ 1.

$$q_{i,i}=-\frac{1}{E\left[V(i)\right]}$$

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Demonstration on How to Calculate the Transition Rate Matrix $Q$

DES - Demonstration on How to Calculate the Transition Rate Matrix Q

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~Example: Relation of $Q$ with the Graphical Representation