[GAP Forum] Find rank-1 matrices in given subspace of matrices

[Dima Pasechnik]

Hi Benoit,
On Mon, Feb 15, 2016 at 11:54:46AM -0500, Benoit Jacob wrote:
> That would give me the set of all rank-1 matrices. I want the set of those
> rank-1 matrices that belong to some given linear subspace of matrices,
> given e.g. as the span of a finite family of matrices.

this is a hard problem. One can write down
a system of polynomial equations specifying the matrices you are interested in.
Let A=sum_k x_k A_k be the matrix of linear forms in variables x_k,
and A_k span your subspace. You need to set all the 2x2 minors of A to 0,
giving you a system of quadratic equations in x_k. Its solutions specify
the rank 1 matrices in your subspace.
It does not seem likely that there is a nice parametrisation of this set.

Hope this helps,
Dima



> Cheers,
> Benoit
> 
> 2016-02-15 11:53 GMT-05:00 Alexander Hulpke <hulpke at math.colostate.edu>:
> 
> >
> > > On Feb 13, 2016, at 4:57 PM, Benoit Jacob <jacob.benoit.1 at gmail.com>
> > wrote:
> > >
> > > Hello,
> > >
> > > What would be a good approach to obtain a parametrization of the set of
> > all
> > > rank-1 matrices in a given subspace of matrices M_n(F), where F is a
> > finite
> > > field?
> >
> > If you want nxm matrices over a field k, why not pick a random nonzero
> > vector v\in k^n and a random normed (i.e. first nonzero coefficient is one)
> > vector w\in k^m and form the (matrix) product v * w^T. I think this gives
> > you a perfect parameterization via parameterizing v and w.
> >
> > Best,
> >
> >    Alexander Hulpke
> >
> > -- Colorado State University, Department of Mathematics,
> > Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
> > email: hulpke at math.colostate.edu, Phone: ++1-970-4914288
> > http://www.math.colostate.edu/~hulpke
> >
> >
> >
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