Re: Re: Questions regarding MatrixExp, and its usage

*To*: mathgroup at smc.vnet.net*Subject*: [mg63381] 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: mathgroup <mathgroup at smc.vnet.net> ><michael_chang86 at hotmail.com> >Subject: [mg63381] Re: [mg63355] Re: [mg63335] Questions regarding MatrixExp, and its >usage >Date: Tue, 27 Dec 2005 15:52:53 +0900 > >*This message was transferred with a trial version of CommuniGate(tm) Pro* > >On 27 Dec 2005, at 09:58, Andrzej Kozlowski wrote: > >> >>On 27 Dec 2005, at 09:42, Andrzej Kozlowski wrote: >> >>> >>>On 27 Dec 2005, at 08:19, Andrzej Kozlowski wrote: >>> >>>> Now you should know that in general this is not going to hold in all >>>>of complex plane (but will hold in most). >>> >>> >>>I wrote the above quite thoughtlessly: obviously there no sense in which >>>the equality "holds in most of the complex plane". Clearly in every >>>sense it holds just as often as it does not. Sorry about that; I replied >>>too quickly. >>>Andrzej Kozlowski >>> >>> >> >> >>One more correction is needed: in a certain obvious sense the equation >>does not hold "more often" than it holds since Exp is a surjective >>mapping of the compelx plane to itself which covers it infinitely many >>times (the fibre is Z - the integers). >> >>Because of the holidays I am now constanlty in a rush and can't find >>enough free time even to write a proper reply! >> >>Andrzej Kozlowski > > >Actually, even the above is not strictly correct: Exp is a surjective >mapping form the complex plane to the complex plane minus the point 0. Now >that I have a little bit of time I can try to analyse the entire problem >more carefully. (In fact, I have not taught complex analysis for over 15 >years and I have become a little bit rusty. So when I first saw this post >I thought the problem lied in the branch discontinuity of Log, which is >why I wrote the relations was true in "most of the complex plane". Of >course I was completely wrong in this respect). > >Let's again define the function > > >f[x_, y_] := E^(x*y) - E^(y*Log[E^x]) > >We want to investigate where in the complex plane this is 0. This is by >definition the same as > > >f[x, y] > > >E^(x*y) - (E^x)^y > > >First, this is going to be zero for any real x and and an arbitrary y: > > >ComplexExpand[f[x,y],{y}] > >0 > >Secondly, suppose we have any pair of complex numbers a,b where f[a,b] ==0. >That is: > >a /: f[a, b] = 0; > >Then we have > > > >ExpandAll[FullSimplify[ > f[a + 2*Pi*I, b]]] > > >E^(a*b + 2*I*b*Pi) - (E^a)^b > > >This will be zero if an only if b is an integer: > > >Simplify[%,Element[b,Integers]] > >0 > >So for every pair (a,b) for which the identity holds and b is not an >integer we can generate uncountably many pairs for which it does not hold >by simply adding 2*Pi*I to a. For example: > > >f[2,3/4] > >0 > > >FullSimplify[f[2 + 2*Pi*I, > 3/4]] > > >(-1 - I)*E^(3/2) > >On the other hand, we can get pairs of complex numbers for which the >identity holds provided the imaginary part of the first complex number is >not large: > > >f[1,2+3I] > > >0 > >However, for complex numbers with large imaginary part: > > >Simplify[f[1 + 12*I, 2 + 3*I]] > >(-E^(-34 + 27*I))* > (-1 + E^(12*Pi)) > >it is easy in this way to give a complete description of the pairs (a,b) >for which f is 0, but I will skip it and turn to matrices. > >In this case, while I am not 100% sure, I tend to believe the situation to >be quite analogous. We are interested in the equation > >MatrixExp[B*p]==MatrixPower[MatrixExp[B],p] Many thanks to Pratik, Daniel, and Andrzej for their very insightful and expert feedback! :) >I believe this will hold for real matrices B and (probably) all complex p >but will not hold in general. In fact I believe most what I wrote above >can be generalised to this case, although the statements and proofs would >be more complicated. Hmm ... actually, from the sample example listed below, I don't believe that it will hold *in general* for real B *and* real p: In[1]: params={theta->Pi^Pi,p->Sqrt[2]}; In[2]: B=theta {{Cot[theta],Csc[theta]},{-Csc[theta],-Cot[theta]}}; In[3]: test1=Simplify[MatrixExp[B p]/.params]; In[4]: test2=Simplify[MatrixPower[MatrixExp[B],p]/.params]; In[5]: Simplify[test1 == test2] Out[5]: False Daniel has suggested that for (square matrix) B and (scalar) p both being real-valued, this only will hold if B is positive definite (although I suspect that this also may hold with B being positive semi-definite too). Using the above example: In[6]: BLim = Limit[B,theta->0]; In[7]: Eigenvalues[BLim] Out[7]: {0, 0} In[8]: MatrixPower[MatrixExp[BLim],p]==MatrixExp[BLim,p] Out[8]: True (By the way ... does anyone know *exactly* what the second argument for MatrixExp does? I've emailed Wolfram, since they only document MatrixExp with one argument, but I've seen their own documentation examples using *two* arguments; empirically, thus far, it seems that: MatrixExp[B,p]==MatrixExp[B p] with p being a scalar. But I digress ...) Anyways, for B and p real, I can 'sorta' see this point from what I (trivially) understand of the Spectral Mapping Theorem, since, as Andrzej has pointed out, Exp[a]^b !=Exp[a b] in general, with a complex, and b real; hence, any (strictly complex-valued) eigenvalue of MatrixPower[MatrixExp[B],p] will in general *not* be equal to MatrixExp[B p] Does this seem reasonable? Overall, too, I guess that I'm still kinda perplexed by what MatrixPower[B,pi] *means*? Somehow, I can feel 'comfortable' with MatrixExp[B pi] but not with MatrixPower[MatrixExp[B],pi] since I tend to think of MatrixExp[B pi] as Limit[MatrixExp[B t],t->Pi] (and can even revert back to an infinite power series matrix sum for an additional 'ease of understanding'), but, unlike general 'x^y' for scalars, can't quite grasp what the MatrixPower[*,*] equivalent signifies ...) :( >Let's just illustrate this in the case of a 2 by 2 random matrix. > > >B=Array[Random[Integer,{1,6}]&,{2,2}] > > >{{6,1},{5,1}} > >Let's take some complex p, e.g. 1+I > >In[65]:= >N[MatrixExp[B*(1 + I)]]==N[MatrixPower[MatrixExp[B],1+I]] > >Out[65]= >True > >To produce a case where the relationship does not hold just imitate the >procedure for complex numbers given above. First we add to 2Pi * times >the identity matrix to B: > >Z = B + 2 Pi*I IdentityMatrix[2]; > >For p take any non-integer number, real or complex: > > >N[MatrixExp[Z*(1/2)]]==N[MatrixPower[MatrixExp[Z],1/2]] > > >False > >Andrzej Kozlowski Do any of my comments above make sense (or does anyone have a better explanation of what exactly how MatrixPower[B,p] can be interpreted with p not an integer)? My musings simply are from a 'layman's' perspective, and probably not very mathematically 'strict' ... :( Regards, Michael