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MathGroup Archive 2003

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Re: Solving a matrix equation

  • To: mathgroup at smc.vnet.net
  • Subject: [mg44340] Re: [mg44266] Solving a matrix equation
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Wed, 5 Nov 2003 10:01:57 -0500 (EST)
  • References: <200311040823.DAA10425@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

josegomez at gmx.net wrote:
> 
> Hi,
>     As a test of my calculations, I would like to solve a matrix
> equation using Mathematica. It is a  symbolic problem, but I can't get
> my head round it.
> 
>     Let T and Q be two nxn complex matrices (T is Hermitian, Q is
> not). I want to test whether a vector p (nx1) is an eigenvector of the
> following combination:
>     A=(Inverse[T]*Q)*(Inverse[T]*Conjugate[Transpose[Q]]),
> or, in LaTeX form:
> (T^{-1}Q)*(T^{-1}*Q^{*T}.
> 
>     I have punched the previous lines into Mathematica, and tried
> to see whether my vector p was an eigenvector. However, I had
> Mathematica eat up all the memory and subsequently crash, so I am
> asking here to see whether someone can suggest a way around this.
> 
>     The problem is 3x3 (n=3), but due to the relatively large number
> of parameters, it is complicated and error-prone to do by hand.
> 
>     (and if it hasn't showed up yet, I am very much a newbie at
> Mathematica!)
> 
>     Many thanks for your time,
>     Jose

It is difficult to say much without specific values for the matrices and
vector. One suggestion I would make is to regroup the dot products so
that you always multiply matrix . vector and never matrix . matrix. This
may lower the intermediate swell. You can also attempt intermediate
simplifications (that is, interposed between dot products); this is
largely heuristic (read: voodoo) and will take trial and error. Also,
the nature of the simplification may depend on what is in the matrices.

Some probabilistic approaches would include substituting numeric values
for parameters, either at the outset or in some intermediate steps.


Daniel Lichtblau
Wolfram Research


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