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

Nils Wagner nwagner@iam.uni-stuttgart...
Wed Jul 11 10:03:23 CDT 2007

```Nils Wagner wrote:
> Anne Archibald wrote:
>
>> On 06/07/07, Nils Wagner <nwagner@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
>> _______________________________________________
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>>
>>
> Hi Anne,
>
> Thank you very much for your detailed information !
> I have enclosed a small script (which is hopefully correct)
> illustrating the computation of the B-orthogonal complement of the null
> space of A for dense matrices.
> How can I extend it to sparse matrices ? I mean what can be used instead
> of linalg.qr and linalg.svd ?
>
> Nils
>
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>
>
Oops, I missed out the attachment.

Nils

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