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Re: Diagonalizing a non-Hermitian Matrix

• To: mathgroup at smc.vnet.net
• Subject: [mg58842] Re: Diagonalizing a non-Hermitian Matrix
• From: "Ituran" <isturan at gmail.com>
• Date: Wed, 20 Jul 2005 00:29:49 -0400 (EDT)
• References: <200507170703.DAA02601@smc.vnet.net> <200507171703.NAA13370@smc.vnet.net> <dbie59$bs4$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

Hi Patrik,

Thanks for your help. As you mentioned SVD is not available in version
4.0.
The matrix that I have picked up as an example  was non-Hermitian and
had complex eigenvalues.
The matrix that I am dealing with is also non-Hermitian but can't have
complex eigenvalues (the diagonal entries indeed correspond to masses).
So, it looks that Mathematica is giving me the right answer.

In order to make the eigenvalues of M real, I should use further  the
phase
arbitrariness  of U,V matrices and pick up the right phases to get real
entries on the diagonal. That is, if I multiply, for example, V by a
unitary phase matrix
P=Diag(E^(i gam1), E^(i gam2)), I have

U.M.(P.V)^(-1)=Diag(Sqrt[14.672],Sqrt[36.328]).

Now, I am solving these two equations for the phases gam1 and gam2.
This way, I am able to get always real values on the diagonal. The same
procedure needs to be repeated for real non-symmetric matrices but
there it is enough to pick up some simple phases like Pi/2,Pi to solve
the (minus) problem on the diagonal. However, here the phases become
non-trivial and depend on the elements of the matrix. That was my
mistake. I though it would still be okay  to choose such simple phases
for complex cases.

I think even if I use JordanDecomposotion, I still need to do the same
trick. As far as I know, there is no such built in function in
Mathematica which gives me the matrices U and V with right phases. Can
SVD do that automatically? 

Best Regards,
I.Turan

• References:
  • Diagonalizing a non-Hermitian Matrix
    • From: "Ituran" <isturan@gmail.com>
  • Re: Diagonalizing a non-Hermitian Matrix
    • From: Pratik Desai <pdesai1@umbc.edu>