Spaces of singular matrices and matroid parity (Q697076)

A subspace \(W\) of the vector space of \(n \times n\) matrices (over a field \(F\)) is called singular if each matrix in it is singular. The main result is that (for \(F\) of characteristic zero) the minimum dimension of a maximal singular subspace \(W\) of the space of skew-symmetric \(2n \times 2n\) matrices is \(3n - 3\). The proof uses the matroid parity theorem of Lovász. It is also proven, using Edmonds' matroid intersection theorem, that if the maximal singular subspace \(W\) of the space of all \(n \times n\) matrices (over an arbitrary field \(F\)) is generated by rank one matrices, then its dimension is at most \(\lfloor{3n^2 - 2n \over 4} \rfloor\), and this number can be achieved. The authors state that the latter result was obtained independently by G.~Kalai.