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, $M=[M_1,\dots ,M_n]$. 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 $m\le p$ such that $[Q{M}_1,\dots ,Q{M}_n]$ is in reduced echelon form and in all but at most $m–1$ blocks $Q{M}_i$ the last $m$ entries of each column are either all zero or they include a non-zero non-unit.