Interpreting the mean first passage matrix of a Markov chain Consider the following first passage matrix:

I just want to know whether one can give a good interpretation to this matrix. All I know to say is that it takes this long to go from this state to that. Is there anything more that can be said? Any help will be much appreciated thanks alot!
 A: Michael Neumann has a nice set of slides on the problem, I borrowed from those slides to answer your question:
Given a Markov process on the state space $\{1,2,\ldots,7\}$ with transition matrix $\mathbf{T}$, $(X_t)_{t\in\mathbb{N}}$, the mean first passage matrix $\mathbf{M}$ is made of entries $m_{ij}$ that are the expected numbers of steps to reach state $j$ when starting with $X_0=i$. So
$$m_{ij}=\sum_{k=1}^\infty k\, \mathbb{P}\left(\min\{\ell\ge 1; X_{\ell}=j \mid| X_0=i\}=k\right)$$ 
So it is the average time it takes the chain to move from $i$ to $j$. For instance, $m_{ii}=1/\pi_i$ where $(\pi_1,\ldots,\pi_7)$ is the stationary distribution of the Markov chain.
The formula for computing the matrix $\mathbf{M}$ has been derived by Carl Meyer, in this 1975 paper.
A: What you can do depends on what kind of data you have. However, in my field, which is the study of thermodynamic systems like proteins, the MFPTs from markov models can be combined with things like Perron cluster analysis (PCCA) or transition pathway analysis (TPT) to help better understand how the system behaves on a more intuitive level.
For example, maybe your data is really just describing the transition between two dominant groups of states (say, the folded and unfolded states of a protein). PCCA would help you cluster those states rigorously and you can use the MFPTs figure out what the transition rates between those states are (in this example, folding times).
Another thing you may want to look at are the fluxes of the system/matrix (see second link above), which could tell you the dominant pathway from one state (source) to another (sink).
This all applies to ergodic systems/matrices, however. I'm not sure how well it transfers to something like absorbing markov models, so be mindful of the type of data you have. Anyway, I hope this leads you in the right direction.
