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# exponential rule application
I'm trying to compute the following exponential of a matrix with
elements that are constants:
A={{0,1},{-1,0}};
MatrixExp[A*t]
The answer can be shown to be {{Cos[t], Sin[t]},{-Sin[t],Cos[t]}} by
using the definition of the matrix exponential and expanding a few
terms in the series. However, MatrixExp[A*t]//Simplify returns the
following expression:
{{(1/2*(1 + E^(2*I*t)))/E^(-(-I*t)),
(-(1/2)*I*(-1 + E^(2*I*t)))/E^(-(-I*t))},
{(1/2*I*(-1 + E^(2*I*t)))/E^(-(-I*t)),
(1/2*(1 + E^(2*I*t)))/E^(-(-I*t))}}
Clearly, we need to apply the simple rule that
complexExpRule=Exp[a_*I*theta_]->Cos[a*theta]+I*Sin[a*theta]
However, I can't seem to make this rule simplify the output of
MatrixExp[A*t]. For example, the following simple expression,
Exp[2*I*t]/.complexExpRule, returns, Exp[2*I*t], instead of,
Cos[2*t]+I*Sin[2*t]. Curiously, Exp[r*I*t]/.complexExpRule, returns,
Cos[r*t]+I*Sin[r*t], as we would hope.
Any help would be appreciated.
John Albert Horst
Intelligent Systems Division
National Institute of Standards and Technology Bldg 220 Rm B-124
Gaithersburg, MD 20899
voice: (301)975-3430
email: john.horst@nist.gov
internet: http://isd.cme.nist.gov/staff/horst/
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