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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
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