Re: Questions regarding MatrixExp, and its usage

*To*: mathgroup at smc.vnet.net*Subject*: [mg63350] Re: [mg63335] Questions regarding MatrixExp, and its usage*From*: Pratik Desai <pdesai1 at umbc.edu>*Date*: Sat, 24 Dec 2005 16:02:58 -0500 (EST)*References*: <BAY105-F26D2B5ABD484AE9CD7EF289A320@phx.gbl>*Sender*: owner-wri-mathgroup at wolfram.com

Michael Chang wrote: > Hi Pratik, > > Many thanks for your response and help! > > My only concern is about the usage of MatrixPower -- all of the > Mathematica online documentation and examples using this function seem > to indicate that it is only valid for an *integer* power p. > > Since MatrixExp[aMatrix,p] exists (and is unique) for all square > "aMatrix" values and any *complex* value of "p", I guess that I began > wondering under what conditions this might be equal to > > MatrixPower[MatrixExp[aMatrix],p] > > ? Perhaps mathematically this only holds for *integer* values of p? > I don't know ... > > Anyways, many thanks again, and Happy Holidays! > > Regards, > > Michael > >> From: Pratik Desai <pdesai1 at umbc.edu> To: mathgroup at smc.vnet.net >> To: "michael_chang86 at hotmail.com" <michael_chang86 at hotmail.com> >> Subject: [mg63350] Re: [mg63335] Questions regarding MatrixExp, and its usage >> Date: Sat, 24 Dec 2005 09:30:52 -0500 >> >> michael_chang86 at hotmail.com wrote: >> >>> Hi, >>> >>> For any arbitrary (possibly complex-valued) square matrix A, >>> Mathematica enables the computation of the matrix exponential of A via >>> >>> In[1]: A={{ some square matrix}}; >>> In[2]: expA=MatrixExp[A]; >>> >>> 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 ...? >>> >>> As usual, any and all help would be greatly appreciated! >>> >>> Regards, >>> >>> Michael >>> >>> >>> >> How about MatrixPower >> matx[A_?MatrixQ, p_]=MatrixPower[MatrixExp[A], p] >> >> >> Hope this helps >> >> Pratik >> >> >> >> > > You will never know unless you try :-) In[10]:= p=Random[]+Pi*I MatrixPower[MatrixExp[IdentityMatrix[3]],p]//Chop//InputForm Out[10]= 0.982433\[InvisibleSpace]+3.14159 \[ImaginaryI] Out[11]//InputForm= {{-2.670947256395083, 0, 0}, {0, -2.670947256395083, 0}, {0, 0, -2.670947256395083}} In[33]:= MatrixExp[IdentityMatrix[3],p]//Chop//InputForm Out[33]//InputForm= {{-2.6709472563950825, 0, 0}, {0, -2.6709472563950825, 0}, {0, 0, -2.6709472563950825}} I think in my experience with mathematica if there are some limitation with a particular function, the documentation always seems to highlight it somewhere, and I did not see any explicit disclaimers regarding the limitation for MatrixPower only working with integers. To be perfectly honest, I don't know why In[33] works perhaps someone else on the forum can help Pratik Happy Holidays to you! PS: I hope you don't mind my posting your reply on the forum -- Pratik Desai