Re: Re: Re: Questions regarding MatrixExp, and its usage

*To*: mathgroup at smc.vnet.net*Subject*: [mg63407] Re: [mg63390] Re: [mg63355] Re: [mg63335] Questions regarding MatrixExp, and its usage*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Fri, 30 Dec 2005 02:32:25 -0500 (EST)*References*: <200512281024.FAA00331@smc.vnet.net> <50213.70.144.61.64.1135788300.squirrel@webmail.wolfram.com>*Sender*: owner-wri-mathgroup at wolfram.com

>>>>> >>>>> 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 Table[KroneckerDelta[i ² j, True] *Derivative[j - i][f][r]/(j - i)!, { i, 1, k}, {j, 1, k}] 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. Andrzej

**References**:**Re: Re: Questions regarding MatrixExp, and its usage***From:*"Michael Chang" <michael_chang86@hotmail.com>