On the matrices of given rank in a large subspace

Details On the matrices of given rank in a large subspace On the matrices of given rank in a large subspace

2010-04-05

arxiv.org/abs/1004.0579v2

Linear Algebra Appl. 435 (2011), 137-141

Mathematics

Rings and Algebras

8 pages

Scientific paper

10.1016/j.laa.2011.01.009

Let V be a linear subspace of M_{n,p}(K) with codimension lesser than n, where K is an arbitrary field and n >=p. In a recent work of the author, it was proven that V is always spanned by its rank p matrices unless n=p=2 and K is isomorphic to F_2. Here, we give a sufficient condition on codim V for V to be spanned by its rank r matrices for a given r between 1 and p-1. This involves a generalization of the Gerstenhaber theorem on linear subspaces of nilpotent matrices.

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