Conditional \(S\)-matrices. (Q1415295)

Conditional \(S\)-matrices and conditonal \(S^*\)-matrices (denoted as \(CS\)-matrices and \(CS^*\)-matrices, respectively), were first analyzed by \textit{R. A. Brualdi}, \textit{K. L. Chavey} and \textit{B. L. Shader} [Conditional sign-solvability, Math. Comput. Modelling 17, No. 1, 141--148 (1993; Zbl 0768.90051)] and \textit{R. A. Brualdi} and \textit{B. L. Shader} [Matrices of sign-solvable linear systems (Cambridge Tracts in Mathematics 116, Cambridge Univ. Press., Cambridge) (1995; Zbl 0833.15002)]. They introduced these matrices as a generalization of the \(S\)-matrices and \(S^*\)-matrices, and they proved that the \(CS\)- and \(CS^*\)-matrices play an important role in the study of conditionally sign solvable linear systems; that is, systems that are solvable and both its solvability and the sign pattern of its solution vector are uniquely determined by the sign patters of the coefficient matrix and terms independent matrix. In this paper, the authors study various properties and recognition criteria of \(CS\)-matrices. More precisely, they present a generalization of some properties of the \(S\)- and \(S^*\)-matrices. They also introduce the sign order of matrices, and the concept of majorized row over a matrix to generalize the conformal contraction operation which preserves the \(CS\) and \(CS^*\) property of matrices (for \(S\)- and \(S^*\)-matrices the generalized conformal contractions are the same as the usual conformal contractions.) In addition, several special classes of \(CS\)-matrices, such as square \(CS\)-matrices, barely \(CS\)-matrices (\(BCS\)-matrices) and maximal \(CS\)-matrices, are analyzed. For \(BCS\)-matrices, the authors give a sharp lower and upper bounds for the number of rows and the number of non-zero entries in terms of the number of columns of the given matrix. They also propose a problem about the largest number of non-zero entries of \(BCS\)-matrices. For maximal \(CS\)-matrices, the authors obtain several necessary conditions and several sufficient conditions for a matrix to be a maximal \(CS\)-matrix. For square \(CS\)-matrices they obtain the conditions for a \(CS\)-matrix to be a \(S\)-matrix. They also use graph theoretical methods to give a characterization and recognition algorithm for square \(CS\)-matrices.