I want to ask a question about absorbing Markov Chain. Let $P=\begin{bmatrix}Q&R\\0&1\end{bmatrix}$ be an absorbing Markov chain with $n$ states. Let $x$ be some unit vector of some initial state. We make the following assumptions:
(1) There is a Hamilton Path (see the Markov Chain as a directed graph) from the initial state, through all the states, to the absorbing state.
(2) $Q$ is diagonalizable, and from any initial state, it is absorbed with $1$ probability.
My question is, is it true that given the distribution of absorbing time, the spectral of $Q$ is unique? Or, given a unique set of spectrals, is it true that the distribution of absorbing time is fixed?
The background of this problem is that we are looking for some expectation of some stopping time problem of some Markov Chain, and after searching for the literature, the expected time is the same as another problem. More importantly, we found that (but not yet proved) the distribution of the stopping time is identical. But when we are trying to solve the problem, it seems that the Markov Chain is nowhere identical. We believe that it is just a change-of-basis of the Markov Chain and we have checked for some small cases where the spectral is identical. So, I have the question from above. We hope (but we don't know whether) looking at the spectral of the Markov Chains helps us to establish the same distribution of expectation of the absorbing time.