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Re: orthonormal eigenvectors]

  • To: mathgroup at smc.vnet.net
  • Subject: [mg88469] Re: orthonormal eigenvectors]
  • From: mante <claude.mante at univmed.fr>
  • Date: Tue, 6 May 2008 06:39:15 -0400 (EDT)

First, since m is not hermitian, its eigenvectors are not necessarily 
orthogonal with each other.

In[17]:= res = Eigenvectors[m]
Out[17]= {{0, 0, 1}, {(-I)*Sqrt[2], 1, 0}, {I/Sqrt[2], 1, 0}}

Relatively to the usual euclidean scalar product, we can see that these 
vectors are not orthogonal :

In[55]:= Outer[Dot, res, res, 1] // N
Out[55]= {{1., 0., 0.}, {0., -1., 2.}, {0., 2., 0.5}}

but notice that Dot is not an hermitian form for *complex vector 
spaces*! One should use instead a scalar product like :

Scal[u_List, v_List] := Dot[u, Conjugate[v]] // ComplexExpand

then we have orthogonality :

In[58]:= Outer[Scal, res, res, 1] // N
Out[58]= {{1., 0., 0.}, {0., 3., 0.}, {0., 0., 1.5}}

Hope this helps...
Claude

-- 
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    Claude Manté

UMR CNRS 6117 LMGEM
http://www.com.univ-mrs.fr/LMGEM/

Centre d'Océanologie de Marseille
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