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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
Abstract

For a given 3 × 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 3 such that xT(A λI)x = 0 and both x and (A λI)x lie in the Lorentz cone, which consists of all vectors in 3 forming a 45 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 M3 of 3 × 3 real matrices, that is, the linear maps ϕ : M3 M3 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 [1])A(QT [1]), where Q is an orthogonal 2 × 2 matrix.

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