Hmmm. Just happened on this abstract from the 1998 ICM (in Berlin): ----- Michaelis Walter J University of New Orleans New Orleans LA U S A The exp onential matrices of noncommuting matrices may com mute We show how to construct entire families of noncommuting by real matrices A and B for which eA eB eAB and yet eA eB eB eA Though it is known that esA etB etB esA s t R A B B A and that eAt eBt eABt t R A B B A our examples provide noncommuting by real matrices A and B for which eAt eB eB eAt t R We also give examples of noncommuting by real matrices and equivalently of by complex matrices A and B such that eA eB eAB I the identity matrix and thus eA eB eAB BA ee ----- Apologies, but I don't have time to fix the screwed-up characters right now. But it claims to find real matrices A,B with AB!= BA, both 2x2 and 4x4, and exp(A)exp(B) = exp(B)exp(A) (no claim about exp(A+B)). If anyone can e-mail me a copy of his paper that would be greatly appreciated. --Dan On Aug 4, 2014, at 6:17 PM, Dan Asimov <dasimov@earthlink.net> wrote:
It's well known and easy to prove that if NxN matrices A and B commute, then
(*) exp(A)exp(B) = exp(A+B) = exp(B)exp(A)
.
But AB = BA is not a necessary condition for (*), as googling will readily reveal.
Does anyone know necessary and sufficient conditions on A and B for (*) to hold?
If not, how about a large class of A and B for which (*) holds despite AB and BA being unequal ?
I think such examples of complex matrices are easier to come by than of real ones, so I'm particularly interested in the case where A and B are real.
--Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun