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Re: Eigensystem[hermitianMatrix] sometimes returns non-orthonormal

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  • Subject: [mg96865] Re: Eigensystem[hermitianMatrix] sometimes returns non-orthonormal
  • From: Jens-Peer Kuska <kuska at>
  • Date: Thu, 26 Feb 2009 07:57:24 -0500 (EST)
  • Organization: Uni Leipzig
  • References: <go31in$fb0$>
  • Reply-to: kuska at


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.


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 (
> ), 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., ).
> 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

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