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MathGroup Archive 2001

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Re: diagonalization

  • To: mathgroup at smc.vnet.net
  • Subject: [mg31357] Re: diagonalization
  • From: gah at math.umd.edu (Garry Helzer)
  • Date: Tue, 30 Oct 2001 04:35:41 -0500 (EST)
  • Organization: University of Maryland
  • References: <9rdfmc$7nk$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

In article <9rdfmc$7nk$1 at smc.vnet.net>, Mitsuhiro Arikawa
<arikawa at mpipks-dresden.mpg.de> 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.

For a hermitian matrix, eigenvectors belonging to different eigenvalues
are orthogonal. If an eigenvalue is repeated then the space of
eigenvectors has dimension >1 and so even independent eigenvectors need
not be orthogonal. 

If an eigenvalue is repeated Mathematica will return a basis of the
eigenspace. If you want an orthogonal basis of the eigenspace, apply the
QR algorithm or Gram-Schmidt process to the eigenvectors provided.


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