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Re: Eigenvectors

• To: mathgroup@smc.vnet.net
• Subject: [mg10286] Re: [mg10275] Eigenvectors
• From: Wouter Meeussen <eu000949@pophost.eunet.be>
• Date: Sat, 3 Jan 1998 23:24:16 -0500

At 05:07 3-01-98 -0500, David Djajaputra wrote:
>Is there a simple command that can give me the <b>normalized</b>
>eigenvectors of a hermitian matrix?
>Or the unitary matrix that diagonalizes it?
>
>Thanks.
>
>David
>
>
this gives you a random hermitian 3.3 matrix:
(m=(w=Table[Random[Complex],{3},{3}])+
Transpose[Conjugate@w])//MatrixForm

these are the eigenvectors:
v=Eigenvectors[m]//Chop
and these are the (real!) eigenvalues: e=Eigenvalues[m]//Chop

the v are already normalised:
(* do not forget to take conjugates! *) so this should give {1.,1.,1.}:
(# . Conjugate@# )&  /@ v    //Chop

and for a hermitian m, they are orthogonal: Transpose[Conjugate[v]] . v 
//Chop
gives :
{{1., 0, 0}, {0, 1., 0}, {0, 0, 1.}}

and the following gives the diagonalised matrix of m:
Inverse[Transpose[v]].m.Transpose[v]//Chop//MatrixForm

wouter.
Dr. Wouter L. J. MEEUSSEN
w.meeussen.vdmcc@vandemoortele.be
eu000949@pophost.eunet.be