Re: Re: Re: Questions regarding MatrixExp, and its usage
- To: mathgroup at smc.vnet.net
- Subject: [mg63564] Re: [mg63390] Re: [mg63355] Re: [mg63335] Questions regarding MatrixExp, and its usage
- From: "Michael Chang" <michael_chang86 at hotmail.com>
- Date: Thu, 5 Jan 2006 03:12:39 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
Hi,
Happy New Year to all! :)
>From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
To: mathgroup at smc.vnet.net
>To: Daniel Lichtblau <danl at wolfram.com>, mathgroup <mathgroup at smc.vnet.net>
>CC: Michael Chang <michael_chang86 at hotmail.com>, Pratik Desai
><pdesai1 at umbc.edu>
>Subject: [mg63564] Re: [mg63390] Re: [mg63355] Re: [mg63335] Questions regarding
>MatrixExp, and its usage
>Date: Thu, 29 Dec 2005 21:05:55 +0900
>
>*This message was transferred with a trial version of CommuniGate(tm) Pro*
>>>>>>
>>>>>>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).
>>
>>What I suggested, or should have in case I misworded it, is that positive
>>definiteness is a sufficient assumption for the identity to hold. It may
>>hold in other cases as well.
>>
>>It will hold if
>>
>>(1) p is integer
>>
>>or
>>
>>(2) B (in notation above) is positive definite
>>
>>(or, I think)
>>
>>(3) -1<=p<=1.
>>
>>A case where it perhaps does not hold (I'm trvelling and on the far side
>>of the WRI firewall, hence cannot check this):
>>
>>p = 3/2, B = {{0,-1}, {1,0}} or perhaps the negative thereof. The idea
>>being to use a matrix that emulates complex multiplication by Sqrt [-1].
>
>Why, of course! When I wrote "will hold for real matrices B" I was just
>being stupid and thinking of "matrices with real eigenvalues". What I
>really meant was normal matrices with real eigenvalues, which I think is
>probably a sufficient condition, since it seems that in this case the
>condition will hold if it holds for the eigenvalues. Is that right?
>
>I was in a great hurry writing all this since out daughter is visiting us
>from Germany for the holidays and we are constantly going out or going on
>trips and I can only spare a few minutes at a time for these messages.
>But let me quickly explain what I meant when I wrote about Jordan
>canonical forms. If my memory does not fail me there is the following
>standard procedure for defining f(A) for A is any complex matrix where
>f[x] is any complex function for which certain derivatives exists. Take
>the Jordan decomposition of A. Let its Jordan canonical form be
>
>A = Inverse[S]. Sum[B_i,{i,1,m}].S
>
>where the B_i are the Jordan blocks. We shall define f[A] as the obvious
>sum once we have defined f[B_i]. So we need only define f [B_i]. Suppose
>the minimal polynomial of B_i is p(x) = (x-r)^k. Then we define f[B_i] as
>the following matrix
Hmm ...
>Table[KroneckerDelta[i ² j, True] *Derivative[j - i][f][r]/(j - i)!, { i,
>1, k}, {j, 1, k}]
One symbol doesn't appear to be quite right on my screen (or perhaps my
understanding fails me!), but in the above line, should this be:
KroneckerDelta[i<=j,True] (a 'funny' symbol seems to appear in my message
between the i and j)
and should the Derivative be with respect to 'x', evaluated at 'r' (maybe
that's what's implied, but I just wanted to make sure that I'm not
mis-interpretating this line as the Derivative with respect to 'r')?
>The important thing is that all the derivatives required to define the
>above matrices should exist!
>If they do f[A] will be well defined. So, if I am right, taking f [x_]:=
>x^z ought to give the definition of power of a matrix for provided all the
>above derivatives are well defined. If z is integer certainly here will be
>no problems, and the definition will agree with the usual one.
Perhaps I do *NOT* understand, but here's an example that I considered with
regards to the above definition -- assume that there is simply one Jordan
block, and that A=B (S=I), with
A = B = {{0,1,0},{0,0,1},{0,0,0}};
Suppose that we desire to find Exp[8 A] (MatrixExp[8 A]); the minimal
polynomial in x is x^3, and for i==j, I can see that the diagonal entries
will be Exp[0]==1, and that Exp[A] should be upper triangular -- but the
upper off-diagonal terms don't seem to be quite right (using the above
formula ... for instance, I don't see how I can get the anticipated '8' and
'8^2/2' terms) ...
Have I erred here?
>Andrzej
Again, my sincere thanks to everyone (Andrzej, Daniel, and Pratik) for all
of your insight and help!
Regards,
Michael
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