A linear algebraic view of partition regular matrices
Let \(A=[a_1, a_2, \dots, a_u]\in {\mathbb Q}^{v\times u}\). We say that \(A\) has the columns condition, written \(A\in CC(m)\), if there exists a partition \(\{I_1,\dots,I_m\}\) of \(\{1,\dots,u\}\) of \(A\), such that for all \(t=1,\dots,m\), \(I_t\neq 0\) and \[ \sum_{i\in I_{t}}a_{i} \in\text{Span}\{a_{j} : j\in I_{1}\cup I_2\cup\cdots\cup I_{t-1}\}. \] The authors investigate the properties of the columns condition, especially for oriented incidence matrices of directed graphs and for sign pattern matrices. They show that the oriented incidence matrix of a graph has the columns condition for some \(m\) if and only if it is strongly connected. They also show that a sign pattern matrix has the columns condition for some \(m\) if and only if it has the columns conditions for all \(m\) if and only if each row is either all zeros or contains both \(+\) and \(-\).
scientific article; zbMATH DE number 5811008