Convex combinations of matrices -- full rank characterization (Q1301303)

The authors give a determinant condition which is necessary and sufficient in order that every convex combination of a set of \(n\times m\) complex matrices \(A_1,\dots,A_{k}\) has rank \(n\). The condition is too complicated to state here, but it involves checking that the determinant of a \((k-1)kn\times(k-1)kn\) matrix in \(k\) variables is never equal to \(0\).