[Bug-gsl] Re: [Help-gsl] error in Schur decomposition of a non-symmetric matrix ?

[Francesco Abbate]

> 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