Date: Nov 28, 2012 5:13 PM
Subject: Re: Matrices of rank at least k
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?