\(P_ *\)-matrices are just sufficient
The classes of sufficient matrices and of \(P_*\)-matrices have recently arisen in connection with the linear complementarity problem (LCP). The class \(SU\) of sufficient matrices was defined by \textit{R. W. Cottle}, \textit{J.-S. Pang} and \textit{V. Venkateswaran} [ibid. 114/115, 231-249 (1989; Zbl 0674.90092)] and the class \(P_*\) by \textit{M. Kojima}, \textit{N. Megiddo}, \textit{T. Noma}, and \textit{A. Yoshise} [Lect. Notes Comput. Sci. 538 (1991; Zbl 0766.90077)]. It is known that \(P_*\)-matrices are sufficient. \textit{S.-M. Guu} and \textit{R. W. Cottle} [Linear Algebra Appl. 223-224, 325-335 (1995; Zbl 0831.15013)] have shown that, for \(2\times 2\) matrices, these classes are identical. On the basis of this result and some computational evidence with matrices of order greater than two they conjectured that \(P_*= SU\). In the present paper the author shows that this really is the case. As an consequence he proves that all that has been proved about \(P_*\) in the above mentioned papers holds for \(SU\) also and all results on \(SU\) are valid for \(P_*\), too. For example, the existing finite tests for sufficient matrices can be used as criteria for membership in \(P_*\).
scientific article; zbMATH DE number 881689