无可逆横截子矩阵环上的列分块矩阵
Column-partitioned matrices over rings without invertible transversal submatrices
摘要：
令任何环上的$p \times q$矩阵$M$的列被划分成$n$块，$M = [M_1, ..., M_n]$。如果具有来自不同块$M_i$的列的$M$的$p \times p$子矩阵都不是可逆的，则存在可逆的$p \times p$矩阵$Q$和正整数$m \leq p$，使得$QM = [QM_1, ..., QM_n]$是简化的梯形形式，并且在几乎所有$m-1$块$QM_i$中，每列的最后一个$m$条目或者全为零或者它们包括非零非单位。
英文摘要：
Let the columns of a $p \times q$ matrix $M$ over any ring be partitioned into $n$ blocks, $M = [M_1, ..., M_n]$. 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 $QM = [QM_1, ..., QM_n]$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
