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Re: Eigensystem[hermitianMatrix] sometimes returns

  • To: mathgroup at smc.vnet.net
  • Subject: [mg96857] Re: [mg96805] Eigensystem[hermitianMatrix] sometimes returns
  • From: DrMajorBob <btreat1 at austin.rr.com>
  • Date: Thu, 26 Feb 2009 07:55:56 -0500 (EST)
  • References: <200902250904.EAA15732@smc.vnet.net>
  • Reply-to: drmajorbob at bigfoot.com

I suppose the simplest thing is

Normalize/@Eigenvalues[m]

or

{First@#,Normalize/@Last@#}&@Eigensystem[m]

Bobby

On Wed, 25 Feb 2009 03:04:51 -0600, Yen Lee Loh <yloh 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!  (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
>
>



-- 
DrMajorBob at longhorns.com


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