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Topic: Matrices of rank at least k
Replies: 12   Last Post: Nov 29, 2012 1:15 PM

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Robin Chapman

Posts: 40
Registered: 10/29/12
Re: Matrices of rank at least k
Posted: Nov 29, 2012 3:08 AM
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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 at-least-k 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).



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