Given two similar, diagonizable square matrices $A$ and $B$ that do not commute, can the Baker-Campbell-Hausdorff formula be simplified exploiting the similarity to obtain a nice expression for $\ln(AB)$? (It's probably not simply $\ln A + \ln B$ due to the lack of commutivity)
Can the Baker-Campbell-Hausdorff formula for $\ln(AB)$ be simplified for similar, diagonizable matrices?
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matrices
logarithms
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1Do you know if commutators of similar matrices have any simple form? If commutators of similar matrices have no simpler form due to being similar, I don't see a way that the BCH simplifies for similar matrices. – 2011-08-16
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0@robjohn: not that I know of. Maybe I was hoping for some non-existent free lunch here... – 2011-08-16
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1I'd be somewhat surprised if there were any significant simplification. What helps is common eigenvectors; I don't see how common eigenvalues would. – 2011-08-17
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0@joriki common Eigenvectors would imply simultaneous diagonalizability so the "usual" logarithm formula $\ln(AB)=\ln A + \ln B$ could be applied, right? But yes, I was hoping similarity would still provide ... something at least – 2011-08-17
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0Yes, that's what I meant; I just wanted to emphasize the common context of those two terms, "similarity" and "simultaneous diagonalizability", one implying the same eigenvalues and the other the same eigenvectors. – 2011-08-17