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Re: Matrices of rank at least k
Posted:
Nov 29, 2012 3:08 AM


On 28/11/2012 20:56, Kaba wrote: > 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?
A matrix has rank at least k iff it has a nonsingular k by k submatrix The set of matrices where that particular submatrix is nonsingular serves as the required open neighbourhood. (It is defined by the nonvanishing of a determinant).



