I probably should have tried that yesterday. Instead. I left out the condition A.B != B.A and just tried to find the general solution for exp(A)exp(B) = exp(A+B) = exp(B)exp(A), and it kept reporting that that was beyond the ability of Solve. So finally I set the lower left elements of A and B to 0, and precomputed the exp() matrices, calling them AA, BB, ABAB, to get: ----- In[26]:= Solve[{AB==ABAB,BA==ABAB},{a,b,d,e,f,h},Reals] Solve::nsmet: This system cannot be solved with the methods available to Solve. h a d a e h a + e b E (E - E ) E (E - E ) f d + h Out[26]= Solve[{{{E , -------------- + --------------}, {0, E }} == a - d e - h a + e d + h a + e (E - E ) (b + f) d + h
{{E , -(-------------------------)}, {0, E }}, True},
-a + d - e + h
{a, b, d, e, f, h}, Reals]
But it's reasonable that there may be no real solutions, since I haven't seen any. Now on to the complex (and quaternion and octonion?) ones. --Dan On Aug 5, 2014, at 7:48 AM, Charles Greathouse <charles.greathouse@case.edu> wrote:
Well, if you believe Mathematica, there are no such 2x2 real matrices:
A = {{a11, a12}, {a21, a22}}; B = {{b11, b12}, {b21, b22}}; Solve[MatrixExp[A].MatrixExp[B](*==MatrixExp[A+B]*)== MatrixExp[B].MatrixExp[A] && A.B != B.A, {a11, a12, a21, a22, b11, b12, b21, b22}, Reals] // Timing
{17.191310, {}}
Charles Greathouse Analyst/Programmer Case Western Reserve University
On Mon, Aug 4, 2014 at 9:17 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
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