eigenvector on singlar matrix

[Jose E. Roman]

On 28/07/2009, Takuya Sekikawa wrote:

> Hi
>
> I have a question about SLEPc.
>
> What are EigenVectors calculated when given matrix is singular?
>
> (ex1)
> for example, matrix like this:
>
>    (1 1 1)
> A = (1 1 1)
>    (1 1 1)
>
> matrix A have 2 eigenvalues, one is 0 (double multiple root),
> and other is 3.
>
> in this case eigenvector related to eigenvalue 0, is
> (z1, z2, -(z1+z2))t (z1, z2 can be any value. i.e. freedom degree is  
> 2)
>
>
> (ex2)
>
> B = (0 1)
>    (0 0)
>
> in this case B's eigenvalue is only 0. (double multiple root)
> but eigenvector has only 1 freedom degree.
> (z1, 0)t
>
> My question is, what will be solution by SLEPc in these case?
>
> Thanks,
> Takuya
> ---------------------------------------------------------------
>  Takuya Sekikawa
>         Mathematical Systems, Inc
>                    sekikawa at msi.co.jp
> ---------------------------------------------------------------

For such small matrices, the computed solution will be the same as the  
one provided by Lapack. If your problem matrices are small, use Lapack  
instead of SLEPc.

For large matrices, if the dimension of the nullspace is small then  
you should have no problems when computing the eigenvectors of zero  
eigenvalues with SLEPc. But if you have a large nullspace then things  
may get problematic - I have not tried this case. Please report any  
problems to the SLEPc maintainance email.

Jose