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Re: Eigensystem[hermitianMatrix]sometimes returns

  • To: mathgroup at smc.vnet.net
  • Subject: [mg96916] Re: Eigensystem[hermitianMatrix]sometimes returns
  • From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
  • Date: Fri, 27 Feb 2009 06:12:38 -0500 (EST)
  • Organization: Uni Leipzig
  • References: <200902250904.EAA15732@smc.vnet.net> <go63gk$pn2$1@smc.vnet.net>
  • Reply-to: kuska at informatik.uni-leipzig.de

Hi,

you want to

Normalize/@Eigenvalues[m]


and not the eigenvectors ??

Regards
   Jens

BTW: What helps the normalization when the
vectors are not orthogonal ?

DrMajorBob wrote:
> 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
>>
>>
> 
> 
> 


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