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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


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