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 <hul...@math.colostate.edu>: > > > > > > On Feb 13, 2016, at 4:57 PM, Benoit Jacob <jacob.benoi...@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: hul...@math.colostate.edu, Phone: ++1-970-4914288 > > http://www.math.colostate.edu/~hulpke > > > > > > > _______________________________________________ > Forum mailing list > Forum@mail.gap-system.org > http://mail.gap-system.org/mailman/listinfo/forum _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum
