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MathGroup Archive 2001

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Re: diagonalization

  • To: mathgroup at smc.vnet.net
  • Subject: [mg31349] Re: diagonalization
  • From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
  • Date: Tue, 30 Oct 2001 04:35:32 -0500 (EST)
  • Organization: Universitaet Leipzig
  • References: <9rdfmc$7nk$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Mitsuhiro Arikawa wrote:
> 
> Hello,
> 
> I have one question about "diagonalization".
> 
> There is a case which the eigenvectors are not orthogonal in helmite matrix
> diagonalization. In principle the eigenvectors in hermite matrix should be
> orthogonal. Such a case may occur in the case there is degeneracy.
> 
> The below program is the example. To make a hermite matrix needs a long
> procedure.(Now this is not so important.) Anyway this procedure makes a
> hermite matrix "h", which depends on real number "t" and "alpha".
> (In this exapmle, I will fix "t=1" and "alpha=0.2")
> This program check hermite property and shows the inner product of the
> eigenvectors. It should be unit matrix but result by mathematica is not.
> 
> I would like to know how to solve this problem and
> the function "Eigensystem" is too blackbox (I do not know what happens in
> the calculation). 

I does what EISPACK does in this case. If you wish orthogonal
vectors for degenerated eigenvalues you have to do it by
your self. 

LinearAlgebra`Orthogonalization`

may help. But it is up to you to choose a linear combination,
that satisfy additional contrains.

> I want to know how to control the accuracy. As far as I
> know, there is no option on that.

Set the precision of the input matrix correct and specify a ZeroTest
to be as accurate a possible ?

Regards
  Jens


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