Maximal extension for linear spaces of real matrices with large rank (Q2784209)

Let \(M^{m\times n}\) be the space of \(m\times n\) real matrices. A subspace \(E\) of \(M^{m\times n}\) satisfies the \(k\)-condition if every nonzero matrix in \(E\) has rank greater than \(k\), and \(E\) is \(k\)-maximal if it is a maximal such subspace. The integer \(\chi(m,n,k)\) is defined to be the smallest dimension of all such \(k\)-maximal subspaces. The author gives a constructive method for finding \(k\)-maximal subspaces, and deduces that \(\chi(m,n,k) \geq(m-k) (n-k)\). More detailed calculations with \(k=1\) show that the lower bound is exact in some cases. For instance, when \(n\) is a power of 2, the fact that \(RP^{n-1}\) cannot be embedded in \(R^{2(n-1)-2}\) leads to the equality \(\chi(n,n,1)=(n-1)^2\).