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