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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
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