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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, no identity, Noah Snyder, tomasz Oct 9 '12 at 21:00

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What did you try? –  Did Oct 7 '12 at 9:23

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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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