[SciPy-user] B-orthogonal complement of the null space of A

[Anne Archibald]

On 06/07/07, Nils Wagner <nwagner at iam.uni-stuttgart.de> wrote:
> Hi all,
>
> Consider  two symmetric matrices A and B, where B is positive definite
> and A is positive semidefinite.
> How can I compute the B-orthogonal complement of the null space of A
> with scipy.linalg ?

Find the nullspace of A - which is numerically tricky, of course -
using eig or svd; the B-orthogonal complement of a set of vectors is
the nullspace of a rectangular matrix VB, where V has a row for each
vector in the nullspace. svd should find that nullspace for you.

This is not a particularly numerically stable operation. I suspect
that the question itself is ill-conditioned, so that no algorithm can
reliably solve it, but there may be a better approach. The svd is
fairly reliable, though, and lets you make your own decisions about
when a linear combination is zero enough.

Anne