Let $Q_1$ and $Q_2$ be the matrices for two continuous Markov chains and suppose there exists an invertible matrix $U$ such that $Q_1=U^{-1}Q_2U$. Show that $$e^{Q_1}=U^{-1}e^{Q_2}U$$
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closed as not constructive by BenjaLim, Did, Norbert, Noah Snyder, tomasz Oct 9 '12 at 21:00
As it currently stands, this question is not a good fit for our Q&A format. We expect answers to be supported by facts, references, or specific expertise, but this question will likely solicit debate, arguments, polling, or extended discussion. If you feel that this question can be improved and possibly reopened, see the FAQ for guidance.
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This has nothing to do with $Q_i$ being Markov chain generators, it follows from the definition of $\exp$ and continuity and linearity of matrix multiplication \[ \exp(Q_1) = \sum_{k=0}^\infty \frac{Q_1^k}{k!} = \sum_{k=0}^\infty U^{-1}\frac{Q_2^k}{k!}U = U^{-1}\cdot \sum_{k=0}^\infty \frac{Q_2^k}{k!}\cdot U = U^{-1}\exp(Q_2)U \] |
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