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Re: orthonormalized eigenvectors

• To: mathgroup at smc.vnet.net
• Subject: [mg44498] Re: orthonormalized eigenvectors
• From: David Wood <me at floyd.attbi.com>
• Date: Wed, 12 Nov 2003 08:01:53 -0500 (EST)
• References: <bonnja$h9h$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

Mahn-Soo Choi <mahn-soo.choi at unibas.ch> wrote:
> As far as I see, the eigenvectors returned from Eigenvectors[] or
> Eigensystems[] are not orthogonal for *Hermitian matrices with
> degenerate eigenvalues*.  (For non-degenate Hermitian matrices, of
> course, the eigenvectors are orthogonal as they should be.)

This is to be expected.

> Of course, I could use the singular value decomposition to
> orthonormalize the eigenvectors.  But then I need to evaluate the
> eigenvectors again to get proper correspondence between the eigenvectors
> and eigenvalues.

Why not use GramSchmidt (see LinearAlgebra`Orthogonalization`) on
what comes right out of Eigensystem? You'll have to slightly redefine 
the inner product if your vectors have complex coefficients, naturally.

> This is very frustrating to me because I have to calculate eigenvalues
> and corresponding *orthonormalized* eigenvectors numerically for quite
> big Hermitian matrices.

> Is there any effecient method working with Mathematica to calculate
> numerically the eigenvalues and corresponding *orthonormalized*
> eigenvectors for Hermitian matrices with possibly *degenerate*
> eigenvalues?

If I remember rightly, Gram-Schmidt takes of order N^3 operations to 
orthonormalize all the eigenvectors of an NxN  matrix.  If you're doing 
*numerics* on big matrices, I'd think Mathematica wouldn't be a great tool.

Good luck.
-- 
David M. Wood, Dept. of Physics, Colorado School of Mines, Golden, CO 80401 
Phone: (303) 273-3853; Fax: (303) 273-3919
http://www.mines.edu/Academic/physics/people/pages/wood.html