Date: Nov 28, 2012 3:56 PM
Author: Kaba
Subject: Matrices of rank at least k


An exercise in a book on smooth manifolds asks me to prove that
(m x n)-matrices (over R) of rank at least k is an open subset of
R^{m x n} (and thus an open submanifold). It is intuitively clear to me
why that is true: an arbitrary small perturbation can add one or more to
the rank of a matrix, but if a matrix is of rank k, then there is a
small open neighborhood in which the rank stays the same. So I should be
able to find a small open neighborhood around each at-least-k rank
matrix which still stays in the set, therefore proving the claim. How do
I find such a neighborhood?