Maximal extension for linear spaces of real matrices with large rank
Kewei Zhang · Proceedings of the Royal Society of Edinburgh Section A Mathematics · 2001
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 ⊥ .