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MathGroup Archive 1998

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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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