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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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