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 ]

Posts: 12,067
Registered: 7/15/05
Re: Matrices of rank at least k
Posted: Nov 28, 2012 4:47 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

Kaba wrote:
>Kaba wrote:
>>but if a matrix is of rank k, then there is a
>>small open neighborhood in which the rank stays the same.

>I mean, does not get lower.

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.


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.