Re: Eigensystem[hermitianMatrix] sometimes returns non-orthonormal

*To*: mathgroup at smc.vnet.net*Subject*: [mg96865] Re: Eigensystem[hermitianMatrix] sometimes returns non-orthonormal*From*: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>*Date*: Thu, 26 Feb 2009 07:57:24 -0500 (EST)*Organization*: Uni Leipzig*References*: <go31in$fb0$1@smc.vnet.net>*Reply-to*: kuska at informatik.uni-leipzig.de

Hi, the online help say "Eigenvectors corresponding to degenerate eigenvalues are chosen to be linearly independent." So we need a example that show you are right. Regards Jens Yen Lee Loh wrote: > Dear Mathematica users, > > In Mathematica 7.0.0 (for Linux), calling Eigenvectors[H] or Eigensystem[H] > for a numerical Hermitian matrix H sometimes returns eigenvectors that are > not orthonormal. > This happens when some eigenvalues are degenerate. (I can supply example > code that illustrates the problem, if necessary.) > > This is not really a bug -- the documentation for Eigensystem[] doesn't make > any guarantees of orthonormality -- but nevertheless it is an annoying part > of Mathematica's design. > This issue has been raised 11 years ago ( > http://forums.wolfram.com/mathgroup/archive/1998/Mar/msg00418.html ), but > that post is corrupted! (Surely Wolfram isn't resorting to censorship?) I > was hoping that in Mathematica 7 I would be able to write something like > > Eigensystem[H, Method->"LAPACK-ZHEEVR"] > > or > > Eigensystem[H, OrthonormalizeEigenvectors->True] > > but no such options seem to exist. One workaround is to apply > Orthogonalize[] to the matrix of eigenvectors, but the documentation for > Orthogonalize[] doesn't guarantee that the orthonormalization will only > occur within the "degenerate subspace". So one has to resort to complicated > fixes (e.g., http://arxiv.org/pdf/hep-ph/9607313 ). > Does anyone have a simpler solution? (For example, is there an easy way to > call LAPACK'S ZHEEVR routine, which guarantees orthornormal eigenvectors, > from Mathematica?) > > Thanks a lot. > Yen Lee Loh > >