Re: Re: Questions regarding MatrixExp, and its usage

*To*: mathgroup at smc.vnet.net*Subject*: [mg63377] Re: [mg63355] Re: [mg63335] Questions regarding MatrixExp, and its usage*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Tue, 27 Dec 2005 04:42:49 -0500 (EST)*References*: <200512250719.CAA01655@smc.vnet.net> <94B75903-6BC7-4E34-83F3-706B65D8A122@mimuw.edu.pl> <43B0286B.5050005@umbc.edu> <835A5DF5-DDB1-4C30-80C1-15895F074328@mimuw.edu.pl> <08253694-9065-4A00-938F-446C89AC7175@mimuw.edu.pl> <B9A2F9B1-4CA7-488B-97B1-129C461D0886@mimuw.edu.pl>*Sender*: owner-wri-mathgroup at wolfram.com

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] 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. 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

**References**:**Re: Questions regarding MatrixExp, and its usage***From:*"Michael Chang" <michael_chang86@hotmail.com>