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 parameter for each state.

I'm struggling to understand a lot of the methods and was hoping someone could show me how this works/walk me through it? I assume I need to start with making sure $Q$ is diagonalizable but I'm not really sure how to express the following;

$P(t) = e^{tQ} = Se^{tD}S^{-1}$

But I've never seen a matrix be in the exponential.. I know $D$ is the diagonal of of eigenvalues but I don't know how to express $e^{tD}$. Thank you in advance for your help!

  • 4
    $\begingroup$ this might be helpful for matrix exponentials in general: en.wikipedia.org/wiki/Matrix_exponential $\endgroup$
    – jld
    Commented Apr 17, 2018 at 2:23
  • $\begingroup$ I added it @Xi'an. Sorry, Math Exchange has something different and I've been told to remove the tag so I got out of the habit. $\endgroup$
    – Nicklovn
    Commented Apr 17, 2018 at 12:40
  • $\begingroup$ @Chaconne Thank you for the source. I'll take a look at that. Wasn't really sure what to search. I kept typing, "matrix in power" and all I got was matrix powers. :/ $\endgroup$
    – Nicklovn
    Commented Apr 17, 2018 at 12:40
  • $\begingroup$ This is explained from an elementary standpoint at stats.stackexchange.com/questions/46389. An example of transition matrix estimation is worked out, in detail, at stats.stackexchange.com/questions/131294/…. $\endgroup$
    – whuber
    Commented Apr 17, 2018 at 13:08