Re: exponential rule application
- To: mathgroup@smc.vnet.net
- Subject: [mg11160] Re: [mg11103] exponential rule application
- From: "C. Woll" <carlw@u.washington.edu>
- Date: Mon, 23 Feb 1998 21:41:20 -0500
Hi John,
Have you tried the command ComplexExpand? Part of the problem that
Mathematica is having is that it doesn't know whether t is real or
complex. ComplexExpand assumes that all parameters are real. Thus,
ComplexExpand[MatrixExp[A t]] //Simplify
returns what you want.
Carl Woll
Dept of Physics
U of Washington
On Sun, 22 Feb 1998, John Albert Horst wrote:
> 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/
>
>