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.