Re: Eigensystem[hermitianMatrix] sometimes returns non-orthonormal

*To*: mathgroup at smc.vnet.net*Subject*: [mg96878] Re: Eigensystem[hermitianMatrix] sometimes returns non-orthonormal*From*: Ray Koopman <koopman at sfu.ca>*Date*: Thu, 26 Feb 2009 07:59:48 -0500 (EST)*References*: <go31in$fb0$1@smc.vnet.net>

On Feb 25, 1:04 am, Yen Lee Loh <y... at mps.ohio-state.edu> 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! An uncorrupted version is at http://groups.google.ca/group/comp.soft-sys.math.mathematica/browse_frm/thread/c105834196d04423 > (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 wa y to > call LAPACK'S ZHEEVR routine, which guarantees orthornormal eigenvectors, > from Mathematica?) > > Thanks a lot. > Yen Lee Loh