Linear maps preserving the Lorentz spectrum of 3 × 3 matrices

Bueno, Maria I.; Faktor, Ben; Kommerell, Rhea; Li, Runze; Veltri, Joey

doi:10.2140/involve.2024.17.121

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Linear maps preserving the Lorentz spectrum of $3 \times 3$ matrices

Maria I. Bueno, Ben Faktor, Rhea Kommerell, Runze Li and Joey Veltri

Vol. 17 (2024), No. 1, 121–152

DOI: 10.2140/involve.2024.17.121

Abstract

For a given $3 \times 3$ real matrix $A$ , the eigenvalue complementarity problem relative to the Lorentz cone consists of finding a real number $λ$ and a nonzero vector $x \in ℝ^{3}$ such that $x^{T} (A - λ I) x = 0$ and both $x$ and $(A - λ I) x$ lie in the Lorentz cone, which consists of all vectors in $ℝ^{3}$ forming a $4 5^{\circ}$ or smaller angle with the positive $z$ -axis. We refer to the set of all solutions $λ$ to this eigenvalue complementarity problem as the Lorentz spectrum of $A$ . Our work concerns the characterization of the linear preservers of the Lorentz spectrum on the space $M_{3}$ of $3 \times 3$ real matrices, that is, the linear maps $ϕ : M_{3} \to M_{3}$ such that the Lorentz spectra of $A$ and $ϕ (A)$ are the same for all $A$ . We have proven that all such linear preservers take the form $ϕ (A) = (Q \oplus [1]) A (Q^{T} \oplus [1])$ , where $Q$ is an orthogonal $2 \times 2$ matrix.

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Keywords

Lorentz cone, Lorentz eigenvalues, linear preservers, $3 \times 3$ matrices

Mathematical Subject Classification

Primary: 15A18, 58C40

Milestones

Received: 31 August 2022

Accepted: 28 January 2023

Published: 15 March 2024

Communicated by Stephan Garcia

Authors

Maria I. Bueno
Department of Mathematics University of California Santa Barbara, CA United States

Ben Faktor
Department of Mathematics University of California Santa Barbara, CA United States

Rhea Kommerell
Department of Mathematics University of California Berkeley, CA United States

Runze Li
Department of Mathematics University of California Santa Barbara, CA United States

Joey Veltri
Department of Mathematics The Pennsylvania State University State College, PA United States

Vol. 17, No. 1, 2024