- 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: <firstname.lastname@example.org>
- 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.