Principal pivot transforms of some classes of matrices
Two classes of matrices are defined, based on principal pivot transforms, and their principal properties are emphasized. For an \(n\times n\) matrix \(A\) the \textit{principal pivot transform} (PPT) \textit{of \(A\) with respect to} \(\alpha\subset\{1,2,\dots,n\}\) is the matrix \(M=\left[\begin{smallmatrix} M_{\alpha\alpha} &M_{\alpha\overline{\alpha}}\\ M_{\overline{\alpha}\alpha} &M_{\overline{\alpha}\overline{\alpha}} \end{smallmatrix}\right]\), where \(M_{\alpha\alpha}=(A_{\alpha\alpha})^{-1},\;M_{\alpha\overline{\alpha}}=-(A_{\alpha\alpha})^{-1}A_{\alpha\overline{\alpha}},\;M_{\overline{\alpha}\alpha}=A_{\overline{\alpha}\alpha}(A_{\alpha\alpha})^{-1},\;M_{\overline{\alpha}\overline{\alpha}}=A_{\overline{\alpha}\overline{\alpha}}- A_{\overline{\alpha}\alpha}(A_{\alpha\alpha})^{-1}A_{\alpha\overline{\alpha}}\), \(\overline{\alpha}=\{1,2,\dots,n\}\setminus\alpha\) and the submatrix \(A_{\alpha\alpha}\) containing the rows and columns in \(\alpha\) of \(A\) is nonsingular. A matrix \(A\) is said to be copositive (\(C_0\)) if \(x^TAx\geq0\;\forall x\in\mathbb{R}^n,\;x\geq0\). \(A\) is almost \(C_0\) if it is not \(C_0\) but all its principal submatrices of order \(n-1\) are \(C_0\). \(A\) is said to be semimonotone (\(E_0\)) if \(\forall y\in\mathbb{R}^n,\;0\neq y\geq0\), \(\exists i\) such that \(y_i>0\) and \((Ay)_i\geq0\). In this paper the class of almost fully copositive (\(C_0^f\)) is defined and studied. A matrix is said to be almost \(C_0^f\) if its PPTs are either \(C_0\) or almost \(C_0\) and there exists at least one PPT of \(A\) which is almost \(C_0\). The second class defined in the paper contains the matrices whose PPTs are either \(E_0\) or almost \(C_0\) and at least one PPT is almost \(C_0\). It is shown that for \(n\geq3\), the matrices of these classes are \(P_0\)-matrices if they have a property connected with the linear complementarity problem. It is stated that these matrices are processable by Lemke's algorithm [see \textit{R. W. Cottle, J.-S. Pang}, and \textit{R. E. Stone}, The linear complementarity problem (1992; Zbl 0757.90078)]. The complete characterization of the class of matrices for which at least one PPT is a \(Z\)-matrix is presented as an open problem.
scientific article; zbMATH DE number 2173027
scientific article