Which of these do you want to learn more about: - What e^A means when A is a matrix - How to think about solutions to linear systems of ordinary differential equations.
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Hmm, I guess not, actually. If that were the case, UHU* + VKV* would be diagonal, with eigenvalues lambda_i + mu_i, if lambda_i are the eigenvalues of H and mu_i are the eigenvalues of K. So the rhs above should be diagonal with e^{i(lambda_i + mu_i)} entries on the diagonal.
That's right. When H and K don't share a common basis there's no obvious reason to expect there to be any relationship between the spectrum of exp(iH) exp(iK) and H and K. Thompson tells you there's a really strong relationship.
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Interesting. Is there a short answer to what U and V are in this case?
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Not as far as I know. In fact, I'm not even certain there's a constructive proof known - unless the situation has changed, Thompson's result depends on the proof of Horn's Conjecture by Allen Knutson, Terry Tao, and Alexander Klyachko, and I've never mastered that.
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