Kaba
Posts:
289
Registered:
5/23/11


Matrices of rank at least k
Posted:
Nov 28, 2012 3:56 PM


Hi,
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 atleastk rank matrix which still stays in the set, therefore proving the claim. How do I find such a neighborhood?
 http://kaba.hilvi.org

