\(S\)-inverse matrices (Q1805324)

An \(S\)-inverse matrix is an inverse-determined matrix having the property that every element in its inverse on or below the main diagonal is negative, whereas every element in the inverse above the main diagonal is positive. The motivation for the investigation of this class comes from recent work in metabolic control analysis. The authors characterize the \(n\times n\) \(S\)-inverse matrices and show how to construct them for \(n \geq 3\). In addition, they also show that an \(S\)-inverse matrix is distinguished among the general class of inverse- determined matrices by the fact that its digraph cannot contain as a subdigraph any member of a well-defined family of digraphs.