*To*: mathgroup@smc.vnet.net*Subject*: [mg11103] exponential rule application*From*: John Albert Horst <john.horst@nist.gov>*Date*: Sun, 22 Feb 1998 14:55:29 -0500

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/