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

[Patrick Alken]

On 03/23/2010 02:50 AM, Francesco Abbate wrote:
> 2010/3/23 Patrick Alken<address@hidden>:
>    
> > Yes, T is not strictly upper triangular, and the standard terminology seems
> > to be "quasi upper triangle" since complex eigenvalues will have a non-zero
> > subdiagonal element in T. So T is upper triangle with some possible non-zero
> > subdiagonal elements corresponding to complex eigenvalues.
> >      
> > > I've made some tests and it seems that the function gsl_eigen_nonsymmv
> > > set the whole matrix A to the Schur form T and not only the upper part.
> > >
> > >        
> > I don't believe the lower elements of A are zero'd out and could contain
> > garbage values. I wouldn't rely on these values being 0, but you could call
> > gsl_linalg_hessenberg_set_zero to force them to be 0.
> >      
> Hi Patrick,
>
> thank you very much for your answer.
>
> You are saying that the lower part of A is not set to 0 but so, what
> about the subdiagonal ? I have to guess the exact meaning of what the
> documentation is saying. Now it seems that "only some elements of the
> subdiagonal are set, those that corresponds to 2x2 diagonal
> submatrices corresponding to complex eigenvalues. The other elements
> of the subdiagonal and lower part of A are not set". Is this correct ?
> I believe that the documentation should be explicit enough so that
> people should not guess the meaning of what is written.
>
> For the other side I've made many empirical tests with my software,
> "GSL Shell", and it all the cases I've tested I've found that the
> whole matrix A is set to T, including the lower part.
>
> Francesco
>    

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.