Maximal extension for linear spaces of real matrices with large rank

Abstract

For every 0 < k < min{m,n} and any linear subspace E of real m × n matrices whose non-zero elements have rank greater than k, we show that there is a maximal extension E_max satisfying the same rank condition, and that the dimension of E_max is not less than (m – k)(n – k). We apply this result to the study of quasiconvex functions defined on the complement E^⊥ of E in the form F(X) = f(P_E⊥(X)), where P_E⊥ is the orthgonal projection to E^⊥.