Criteria for sufficient matrices
type: Item
instance of: scholarly article
title: Criteria for sufficient matrices
zbMATH Open document ID: 0857.15019
DOI: 10.1016/0024-3795(94)00058-1
published in: Linear Algebra and its Applications
publication date: 1996-02-14T00:00:00Z
review text: The author proposes new criteria for identifying column sufficient, row sufficient and sufficient matrices. A matrix \(A\in\mathbb{R}^{n\times n}\) is column sufficient if for all \(x \in\mathbb{R}^n\), \(x^*(Ax) \leq 0 \Rightarrow x^*(Ax)=0\) where \(*\) denotes the Hadamard product. If \(A^T\) is column sufficient, \(A\) is row sufficient and \(A\) is sufficient if it is both column and row sufficients. It is proved that if a column (row) sufficient matrix \(A\) is of rank \(r\) and some collection of \(r\) columns (rows) are linearly independent, then the corresponding principal submatrix is nonsingular. As a consequence, if some columns (rows) of \(A\) are linearly independent, then so are the corresponding rows (columns). Based on principal pivoting, the paper gives some new criteria for (column, row) sufficient matrices. A matrix \(A\in\mathbb{R}^{n \times n}\) of rank \(r<n\) is (column, row) sufficient if it is (column, row) sufficient of order \(r+1\). A matrix \(A \in\mathbb{R}^{n \times n}\) with positive determinant is (column, row) sufficient if it is (column, row) sufficient of order \(k<n\) and \(A^{-1}\) is (column, row) sufficient of order \(n-k\). Necessary and sufficient conditions for a (column, row) sufficient matrix \(A\in\mathbb{R}^{n \times n}\) of order \(n-1\) to be (column, row) sufficient are given. Using these conditions, inductive criteria are proposed for (column, row) sufficiency, that is, sequentially check for \(k=2,3, \dots\) whether (column, row) sufficient \(A\) (or \(A^{-1})\) of order \(k-1\) is (column, row) sufficient of order \(k\).
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 90C33
Mathematics Subject Classification ID: 15A03
zbMATH DE Number: 841741
zbMATH Keywords: sufficient matrix
zbMATH Keywords: linear complementarity problem
zbMATH Keywords: \(P\)-matrix
zbMATH Keywords: Hadamard product
zbMATH Keywords: principal pivoting
zbMATH Keywords: rank
author: Hannu Väliaho
reviewed by: Cheng-Xian Xu
MaRDI profile type: Publication
full work available at URL: 0024-3795(94)00058-1
OpenAlex ID: W1982002079
cites work: Duality Theory of Linear Programs: A Constructive Approach with Applications
cites work: The principal pivoting method revisited
cites work: Least-Index Resolution of Degeneracy in Linear Complementarity Problems with Sufficient Matrices
cites work: Complementary pivot theory of mathematical programming
cites work: Two characterizations of sufficient matrices
cites work: Sufficient matrices and the linear complementarity problem
cites work: On the uniqueness of solutions to linear complementarity problems
cites work: On inverting partitioned matrices
cites work: Some generalizations of positive definiteness and monotonicity
cites work: Q3254327
cites work: The linear complementarity problem, sufficient matrices, and the criss- cross method
cites work: Bimatrix Equilibrium Points and Mathematical Programming
cites work: Q4050723
cites work: A combinatorial approach to convex quadratic programming
cites work: Q5183648
cites work: Q3274171
cites work: A procedure for the one-parametric linear complementarity problem
description: scientific article; zbMATH DE number 841741