Re: [eigen] Computing left nullspace of A when rows > cols

[W. James MacLean <james.maclean@xxxxxxxxxxx>]

Thanks Benoit!

left_nullspace(A) = span({x | x^T A = 0}), where A is n by m, and x is an n
by 1 vector.

If you compute an SVD of A in Matlab, for example, the U is an n by n
orthogonal matrix, and V is an m by m orthogonal matrix. The last n-m
columns of U provide a basis for the left nullspace.

In the case where n > m, the contents of the last n-m columns of U multiply
into the 0 rows of the (nominally) n by m S matrix, and thus disappear. So
while you don't need the last n-m columns of U to make the product U S V^*
work, not having them means you don't get a basis for the left nullspace of
A.

Does that describe it well enough?

Cheers,

James

On Fri, 30 Jan 2009 17:38:22 +0100, Benoit Jacob <jacob.benoit.1@xxxxxxxxx>
wrote:
> Hi,
> 
> The SVD is undergoing a rewrite from scratch for 2.1 so it is a very
> good moment to make feature requests.
> 
> Can you please tell me what "left nullspace" means ? I'm a "pure"
> mathematician so I'm not always aware of the matrix computations
> vocabulary.
> 
> I wouldn't spend time modifying the existing SVD, instead Keir and me
> are going to write a new one as soon as we have time.
> 
> Cheers,
> Benoit