An alternative theorem for quadratic forms and extensions
type: Item
instance of: scholarly article
title: An alternative theorem for quadratic forms and extensions
zbMATH Open document ID: 0837.15022
DOI: 10.1016/0024-3795(93)00082-B
published in: Linear Algebra and its Applications
publication date: 1995-04-04T00:00:00Z
review text: This note concerns variations on the following theorem of \textit{Y. Yuan} [Math. Program., Ser., A 47, No. 1, 53-63 (1990; Zbl 0711.90062)]. Theorem: For two real symmetric matrices \(A_1\) and \(A_2\) of order \(n\) by \(n\) the following statements are equivalent: \[ \max \bigl\{ \langle x, A_1x \rangle, \langle x, A_2x \rangle \bigr\} \geq 0 \text{ for all } x \in\mathbb{R}^n; \tag{a} \] (b) there exist \(t_1 \in\mathbb{R}_+\) and \(t_2 \in\mathbb{R}_+\) with \(t_1 + t_2 = 1\), such that the matrix is positive semidefinite. Different extensions of this theorem are discussed, but it is also noticed that the theorem does not extend to the case of more than two quadratic forms. The clearly arranged proof uses convex geometry in arbitrary real topological linear spaces and helps to understand the geometric aspect hidden in the above theorem. Potential applications of the above theorem in other areas are announced but not explored in detail.
reviewed by: Hans-Jochen Bartels
Mathematics Subject Classification ID: 15A63
zbMATH DE Number: 740257
zbMATH Keywords: positive semidefinite
zbMATH Keywords: quadratic forms
zbMATH Keywords: convex geometry
author: J.-P. Crouzeix
author: Juan-Enrique Martínez-Legaz
author: Alberto Seeger
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 740257