The Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Math Forum » Discussions » sci.math.* » sci.math

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Robin Chapman

Posts: 40
Registered: 10/29/12
Re: Matrices of rank at least k
Posted: Nov 29, 2012 3:08 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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).

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2018. All Rights Reserved.