quasi
Posts:
9,097
Registered:
7/15/05
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Re: Matrices of rank at least k
Posted:
Nov 28, 2012 4:47 PM
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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.
quasi
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