Re: Re: Questions regarding MatrixExp, and its usage

*To*: mathgroup at smc.vnet.net*Subject*: [mg63380] Re: [mg63355] Re: [mg63335] Questions regarding MatrixExp, and its usage*From*: "Michael Chang" <michael_chang86 at hotmail.com>*Date*: Wed, 28 Dec 2005 03:55:37 -0500 (EST)*Sender*: owner-wri-mathgroup at wolfram.com

Hi, >From: Andrzej Kozlowski <akoz at mimuw.edu.pl> To: mathgroup at smc.vnet.net >To: Michael Chang <michael_chang86 at hotmail.com> >Subject: [mg63380] Re: [mg63355] Re: [mg63335] Questions regarding MatrixExp, and its >usage >Date: Mon, 26 Dec 2005 23:47:09 +0900 > >*This message was transferred with a trial version of CommuniGate(tm) Pro* > >On 25 Dec 2005, at 16:19, Michael Chang wrote: > >>>>>>I was therefore wondering if >>>>>> >>>>>>MatrixExp[A p]==(MatrixExp[A]^p) >>>>>> >>>>>>where 'p' is an arbitrary complex number, and the '^' operator is my >>>>>>attempt to denote the matrix power, and *not* an element-by- element >>>>>>power for each individual matrix entry. Or does such an expression >>>>>>only hold for real-valued square A matrices? Or am I completely lost >>>>>>here ...? > > >This can't possibly be true for arbitrary square complex matrices since it >is not even true for matrices of dimension 1, that is complex numbers. >In other words, it is not true that Exp[a p]== Exp[a]^p, were a and p are >arbitrary complex numbers. In fact, Mathematica alone can find for you >an example where this is not true. To see that let's define a function f >of two variables: > > >f[a_, b_] := Exp[a*b] - Exp[a]^b > > >If the identity you held for all complex numbers f would have to be >identically zero. However, we can get Mathematica to find an example when >it is not: > > >FindInstance[f[a, b] != 0, {a, b}] > > >{{a -> -(47/10) + (181*I)/10, > b -> 91/10 + (122*I)/5}} > >Since this seems a little hard to verify without Mathematica and since >mathematica is sometimes wrong ;-) it may be more convincing to construct >an example by hand. In fact it is pretty easy. All you need is the well >known identity: > >Exp[I *A] = Cos[A]+ I *Sin[A] Sorry for my tardy response ... I'm trying to catch up on my email! Just a quick typo correction ... I believe that the first "A" definition below should be A=-Pi, given the definition of Exp[I*A] above ... Regardless, the key point that Andrzej has made for complex-valued (square) matrices is duly noted! >Put A = -I*Pi. We get Exp[-I*Pi] = -1 > >Put A = -Pi/2. We get Exp[-Pi/2*I]= -I > >But now note that: > >Exp[-Pi/2*I] == Exp[-Pi*I *(1/2)] > >so if the identity is true than -I == Exp[-Pi/2*I] == Exp[-Pi*I]^ (1/2) = >(-1)^(1/2) == I. > >Thus we get a contradiction. > >Andrzej Kozlowski Regards, Michael