Full Problem: A continuous-time Markov chain has generator matrix
$$Q= \begin{pmatrix} -1 & 1 & 0 \\ 1 & -2 & 1 \\ 2 & 2 & -4 \\ \end{pmatrix} $$
(i) Exhibit the transition matrix of the embedded Markov chain.
(ii) Exhibit the holding time parameters for each state.
OK, I must be misunderstanding something. I have the following for the embedded chain transition probabilities:
$$P_{ij} = \frac{q_{ij}}{q_i}$$
where $q_i$ is the parameters of the exponential length of time that the process stays in i.
The problem is that I know $q_i$ is the minimum with exponential distribution and parameter $\sum_k q_{ik}$.
But when I calculate $P_{11}$ that gives me $-1$ which is impossible..
Obviously there is an error in my thinking but can anyone shine light on what I'm misunderstanding? Thank you greatly in advance.