# Can the Baker-Campbell-Hausdorff formula for $\ln(AB)$ be simplified for similar, diagonizable matrices?

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)

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Do 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. –  robjohn Aug 16 '11 at 15:45
@robjohn: not that I know of. Maybe I was hoping for some non-existent free lunch here... –  Tobias Kienzler Aug 16 '11 at 15:51
I'd be somewhat surprised if there were any significant simplification. What helps is common eigenvectors; I don't see how common eigenvalues would. –  joriki Aug 17 '11 at 5:52
@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 –  Tobias Kienzler Aug 17 '11 at 6:04
Yes, 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. –  joriki Aug 17 '11 at 6:54