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Re: Re: Questions regarding MatrixExp, and its usage
*To*: mathgroup at smc.vnet.net
*Subject*: [mg63391] Re: [mg63355] Re: [mg63335] Questions regarding MatrixExp, and its usage
*From*: "Michael Chang" <michael_chang86 at hotmail.com>
*Date*: Wed, 28 Dec 2005 05:24:13 -0500 (EST)
*Sender*: owner-wri-mathgroup at wolfram.com
Hi Andrzej,
>From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
To: mathgroup at smc.vnet.net
>Subject: [mg63391] Re: [mg63355] Re: [mg63335] Questions regarding MatrixExp, and its
>usage
>Date: Wed, 28 Dec 2005 06:51:40 +0900
>
>
>On 28 Dec 2005, at 01:09, Michael Chang wrote:
>
>>Hi,
>>
>>>From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
To: mathgroup at smc.vnet.net
>>><michael_chang86 at hotmail.com>
>>>Subject: [mg63391] 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
>>
>>
>
>
>You are right of course. I was much too quick optimistic to claim that it
>would hold for all real matrices. Without giving this much thought, I
>imagined that this can be reduced just to the result holding for the
>eigenvalues which is, at best, only the case for certain types of
>matrices; in particular real normal ones.
>I imagine that MatrixPower for arbitrary matrix and arbitrary exponent
>is defined via the Jordan decomposition, by first you defining it for
>Jordan blocks and then taking the suitable sum and finally applying the
>similarity matrices. I have not considered carefully what happens for a
>single Jordan block matrix, but I think the power matrix will have powers
>of the eigenvalues on the diagonal and lower powers of the eigenvalue
>given by differentiation z^p (where p is our exponent) with certain
>coefficients above the diagonal (essentially terms of the "Taylor
>expansion" of z^p). If I remember correctly, this is how one defines f[A]
>for an arbitrary smooth function f and an arbitrary complex matrix A.
>Thinking of this definition, however, I can see no reason, why it should
>be "well behaved" in this case for non-diagonalizable matrices, so I
>suspect real normal matrices (perhaps without zero eigenvalues) are the
>best you can expect in general -although I am not going to have any time
>to check this until well after the New Year.
>
>Wishing everyone a Happy New Year.
>
>Andrzej Kozlowski
Once again, your mathematical expertise and feedback are greatly
appreciated!
For an arbitrary square B matrix, and taking a JordanDecomposition of B,
such that
s.J.Inverse[s]==B,
I can see that
MatrixPower[B,p] == (s.J.Inverse[s] multiplied p times) == s.(J multiplied p
times).Inverse[s]
where p is a (positive) integer, say. But, to my confusion, what does one
obtain when p=Sqrt(Pi), say? For instance, do we still somehow get
multiples of Inverse[s].s==IdentityMatrix[n] in the resulting expression,
somehow?
Anyways, my sincere thanks again for all of your help, and best wishes to
everyone for a Happy New Year! Bonne année! ;)
Regards,
Michael
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