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[Bug-gsl] Re: [Help-gsl] error in Schur decomposition of a non-symmetric
From: |
Francesco Abbate |
Subject: |
[Bug-gsl] Re: [Help-gsl] error in Schur decomposition of a non-symmetric matrix ? |
Date: |
Tue, 23 Mar 2010 16:54:54 +0100 |
> Perhaps a more clear explanation of the Schur form is this: if you take a
> given 2x2 block on the diagonal of the matrix, call it
>
> [ T11 T12 ]
> [ T21 T22 ]
>
> then either T21 = 0, or T21 is nonzero.
>
> If T21 is zero, then T11 and T22 are both real eigenvalues of the original
> matrix.
>
> If T21 is nonzero, then T11 = T22, and T11 +/- sqrt(|T21*T12|) are complex
> conjugate eigenvalues.
>
> So, the subdiagonal element (T21) is guaranteed to be 0 in the case of real
> eigenvalues, and nonzero in the case of complex eigenvalues. I will try to
> update the documentation to make this a little more clear.
>
> The rest of A (below the subdiagonal) is not guaranteed to be 0, as far as I
> remember.
Thank you very much Patrick, now it is completely clear. Well,
actually this is more or less what I have guessed but I was not sure
especially about the subdiagonal and the lower part of the matrix.
Actually I was thinking that for complex eigenvalues the submatrix was
something like:
[ u v]
[ -v u]
à la Cauchy-Riemann but it was a wrong guess... :-)
Otherwise only the problem of the orthogonality of Z remains open...
maybe it is due to the balancing of the matrix made (maybe) internally
by the procedure...
In any case, thank you very much for your help!
Francesco
- [Bug-gsl] error in Schur decomposition of a non-symmetric matrix ?, Francesco Abbate, 2010/03/22
- [Bug-gsl] Re: [Help-gsl] error in Schur decomposition of a non-symmetric matrix ?, Patrick Alken, 2010/03/23
- Message not available
- [Bug-gsl] Re: [Help-gsl] error in Schur decomposition of a non-symmetric matrix ?, Patrick Alken, 2010/03/23
- Re: [Bug-gsl] Re: [Help-gsl] error in Schur decomposition of a non-symmetric matrix ?, Patrick Alken, 2010/03/23
- [Bug-gsl] Re: [Help-gsl] error in Schur decomposition of a non-symmetric matrix ?,
Francesco Abbate <=
- Re: [Bug-gsl] Re: [Help-gsl] error in Schur decomposition of a non-symmetric matrix ?, Patrick Alken, 2010/03/23
- Re: [Bug-gsl] Re: [Help-gsl] error in Schur decomposition of a non-symmetric matrix ?, Francesco Abbate, 2010/03/23
- Re: [Bug-gsl] Re: [Help-gsl] error in Schur decomposition of a non-symmetric matrix ?, Patrick Alken, 2010/03/23
[Bug-gsl] Re: [Help-gsl] error in Schur decomposition of a non-symmetric matrix ?, Leo Razoumov, 2010/03/24