Principal pivot transforms of some classes of matrices
type: Item
instance of: scholarly article
title: Principal pivot transforms of some classes of matrices
zbMATH Open document ID: 1077.15016
author: S. K. Neogy
author: A. K. Das
published in: Linear Algebra and its Applications
publication date: 2005-06-01T00:00:00Z
review text: 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…
reviewed by: Valeriu Prepeliţă
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 15A24
zbMATH DE Number: 2173027
zbMATH Keywords: principal pivot transform
zbMATH Keywords: almost copositive matrix
zbMATH Keywords: semimonotone matrix
zbMATH Keywords: \(Q_{0}\)-matrix
zbMATH Keywords: \(Z\)-matrix
zbMATH Keywords: linear complementarity problem
zbMATH Keywords: Lemke's algorithm
MaRDI profile type: Publication
full work available at URL: j.laa.2004.11.023
OpenAlex ID: W2071866591
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DOI: 10.1016/J.LAA.2004.11.023
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description: scientific article; zbMATH DE number 2173027