Linear maps preserving the Lorentz-cone spectrum in certain subspaces of \(M_n\)
type: Item
instance of: scholarly article
title: Linear maps preserving the Lorentz-cone spectrum in certain subspaces of \(M_n\)
zbMATH Open document ID: 1472.15039
author: Yanyan Li
published in: Banach Journal of Mathematical Analysis
publication date: 2021-07-29T00:00:00Z
review text: Let \(n \geq 3\) and consider the Lorentz cone \(\mathcal K = \{(x,x_n)\in \mathbb R^{n-1}\times \mathbb R : \lVert x \rVert \leq x_n \}\), where \(\lVert x\rVert\) is the 2-norm of \(x\). It is well-known that the Lorentz cone \(\mathcal K\) is self-dual, and so the eigenvalue complementarity problem associated with \(\mathcal K\) is, for a given real-valued \(n \times n\) matrix \(A\), finding a scalar \(\lambda \in \mathbb R\) and nonzero vector \(x \in \mathbb R^n\) such that \[x \in \mathcal K, \qquad (A-\lambda I_n)x \in \mathcal K, \qquad x^T(A-\lambda I_n)x = 0,\] where \(I_n\) denotes the \(n \times n\) matrix and \(x^T\) denotes the transpose of \(x\). The scalar \(\lambda\) is called \textit{Lorentz eigenvalue} of \(A\) and the set of all Lorentz eigenvalues of \(A\) is denoted \(\sigma_\mathcal{K}(A)\), and called the \textit{Lorentz cone spectrum of \(A\)}. The authors explore linear preserver problems related to the Lorentz cone spectrum of matrices. In other words, they obtain characterizations of linear maps \(\phi: \mathcal M \to \mathcal M\) such that \(\sigma_\mathcal{K}(\phi(A)) = \sigma_\mathcal{K}(A)\) for all \(A \in \mathcal M\), where \(\mathcal M\) is a given subspace of \(M_n(\mathbb R)\). The authors call maps of the form \(\phi(A) = PAQ\) or \(\phi(A) = PA^TQ\), where \(P\) and \(Q\) are fixed matrices, \textit{standard linear maps} (most linear preserver problems in the literature conclude that the maps have this particular form; see [\textit{C.-K. Li} and \textit{S. Pierce}, Am. Math. Mon. 108, No. 7, 591--605 (2001; Zbl 0991.15001)] for a general description). Lorentz cone spectrum preservers are considered on the following subspaces \(\mathcal M\): diagonal matrices, block-diagonal matrices of the form \(\tilde{A}\oplus[a]\) with \(a \in \mathbb{R}\) and \(\tilde{A}\in M_{n-1}(\mathbb{R})\) symmetric, block-diagonal matrices of the form \(\tilde{A}\oplus[a]\) with \(a \in \mathbb{R}\) and \(\tilde{A}\in M_{n-1}(\mathbb{R})\) arbitrar…
reviewed by: Hayden Julius
Mathematics Subject Classification ID: 15A86
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 58C40
zbMATH DE Number: 7376337
zbMATH Keywords: Lorentz cone
zbMATH Keywords: Lorentz eigenvalues
zbMATH Keywords: linear preserver
MaRDI profile type: Publication
full work available at URL: s43037-021-00140-y
OpenAlex ID: W3175545913
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DOI: 10.1007/S43037-021-00140-Y
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description: scientific article; zbMATH DE number 7376337