On a hierarchy of generalized diagonal dominance properties for complex matrices

This article concerns dominance conditions for an $n \times m$ matrix. In the simplest kind of dominance, the (absolute) value of the diagonal element exceeds the sum of the absolute values of the nondiagonal elements on the same row. This condition has been generalized in the literature in several ways, of which we consider ways in which the rows of the matrix cooperate. Our work amounts to a sorting out of certain dominance conditions that belong to a class $\mathcal {C}$ of dominance conditions. We prove a theorem characterizing all true statements of the form \[ {C_1},{C_2}, \cdots ,{C_s} \Rightarrow {C_0}\] where ${C_i} \in \mathcal {C}(i = 0,1, \cdots ,s)$.