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Re: Product command with matrices

  • To: mathgroup at smc.vnet.net
  • Subject: [mg87922] Re: Product command with matrices
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Sat, 19 Apr 2008 23:53:36 -0400 (EDT)
  • Organization: The Open University, Milton Keynes, UK
  • References: <fu9ft7$ce2$1@smc.vnet.net>

J Davis wrote:

> I want to define a matrix valued function such as the following (in
> LaTeX lingo):
> 
> $$
> X(0)=Id,
> X(n)=\prod_{i=0}^{n-1} (Id + f[i] A)
> $$
> 
> where A and f have already been defined and Id is the identity matrix
> of appropriate size.
> 
> I tried the following:
> 
> Id=IdentityMatrix[2];
> Phi[0]:=Id;
> Phi[n_]:= Product[Id + f[i] A,{i,0,n-1}]
> 
> However, Phi[3] and (Id+f[2]A).(Id+f[1]A).(Id+f[0]A) do not agree.

The following will do the recursion and correctly handle the *matrix* 
product (which Product[] does not do).

     Fold[(Id + f[#2] A).#1 &, Id + f[0] A, Range[n - 1]]

Here is an example of a complete definition:


     Id = IdentityMatrix[2];

     phi[0] := Id;
     phi[n_] := Fold[(Id + f[#2] A).#1 &, Id + f[0] A, Range[n - 1]]

     phi[3]
     % == (Id + f[2] A).(Id + f[1] A).(Id + f[0] A) // Simplify


     {{A (A (1 + A f[0]) f[1] + A f[0] (1 + A f[1])) f[
     2] + (A^2 f[0] f[1] + (1 + A f[0]) (1 + A f[1])) (1 + A f[2]),
     A (A^2 f[0] f[1] + (1 + A f[0]) (1 + A f[1])) f[
     2] + (A (1 + A f[0]) f[1] + A f[0] (1 + A f[1])) (1 +
     A f[2])}, {A (A^2 f[0] f[1] + (1 + A f[0]) (1 + A f[1])) f[
     2] + (A (1 + A f[0]) f[1] + A f[0] (1 + A f[1])) (1 + A f[2]),
     A (A (1 + A f[0]) f[1] + A f[0] (1 + A f[1])) f[
     2] + (A^2 f[0] f[1] + (1 + A f[0]) (1 + A f[1])) (1 + A f[2])}}

     True


Regards,
-- Jean-Marc


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