Column-Partitioned Matrices Over Rings Without Invertible Transversal Submatrices

Abstract

Let the columns of a \(p \times q\) matrix \(M\) over any ring be partitioned into \(n\) blocks, \(M = [M_1, \ldots, M_n]\). If no \(p \times p\) submatrix of \(M\) with columns from distinct blocks \(M_{i}\) is invertible, then there is an invertible \(p \times p\) matrix \(Q\) and a positive integer \(m \leq p\) such that \([QM_1, \ldots, QM_n]\) is in reduced echelon form and in all but at most \(m – 1\) blocks \(QM_i\) the last \(m\) entries of each column are either all zero or they include a non-zero non-unit.