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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!

Nicklovn
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    this might be helpful for matrix exponentials in general: https://en.wikipedia.org/wiki/Matrix_exponential – jld Apr 17 '18 at 02:23
  • 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. – Nicklovn Apr 17 '18 at 12:40
  • @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. :/ – Nicklovn Apr 17 '18 at 12:40
  • This is explained from an elementary standpoint at https://stats.stackexchange.com/questions/46389. An example of transition matrix estimation is worked out, in detail, at https://stats.stackexchange.com/questions/131294/how-to-estimate-the-infinitesimal-generator-of-a-markov-chain/131325#131325. – whuber Apr 17 '18 at 13:08

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