Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

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


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

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 ]
Kaba

Posts: 289
Registered: 5/23/11
Re: Matrices of rank at least k
Posted: Nov 28, 2012 5:13 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

28.11.2012 23:47, quasi wrote:
> Let m,n be positive integers, and let k be an integer with
> 0 <= k <= min(m,n). The set T_k of m x n matrices of
> rank <= k is easily seen to be closed since, for each k,
> there is a polynomial P_k in m*n variables with real
> coefficients such that an m x n matrix A with real entries
> satisfies the condition rank(A) <= k iff the coefficients
> of A satisfy P_k = 0. Regarding P_k as a function from
> R^(mxn) to R, P_k is continuous, hence ((P_k)^(-1))(0)
> is closed. It follows that T_k is closed for all k. In
> particular, for each k, T_(k-1) is closed, and thus,
> the set of matrices with rank >= k is open.


Sounds good. But how to prove the existence of the polynomials P_k?

--
http://kaba.hilvi.org



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

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.