Re: Re: Questions regarding MatrixExp, and its usage
- To: mathgroup at smc.vnet.net
- Subject: [mg63386] Re: [mg63355] Re: [mg63335] Questions regarding MatrixExp, and its usage
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Wed, 28 Dec 2005 03:55:42 -0500 (EST)
- References: <BAY105-F241B75C5EDB643DA783A549A370@phx.gbl>
- Sender: owner-wri-mathgroup at wolfram.com
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
>> To: mathgroup <mathgroup at smc.vnet.net>
>> CC: Pratik Desai <pdesai1 at umbc.edu>, Michael Chang
>> <michael_chang86 at hotmail.com>
>> Subject: [mg63386] Re: [mg63355] Re: [mg63335] Questions regarding
>> MatrixExp, and its usage
>> Date: Tue, 27 Dec 2005 15:52:53 +0900
>>
>>
>> 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