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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 $\lambda$ and a nonzero vector $x\in {ℝ}^{3}$ such that ${x}^{T}\left(A-\lambda I\right)x=0$ and both $x$ and $\left(A-\lambda I\right)x$ 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 $\lambda$ 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 $3×3$ real matrices, that is, the linear maps $\varphi :{M}_{3}\to {M}_{3}$ such that the Lorentz spectra of $A$ and $\varphi \left(A\right)$ are the same for all $A$. We have proven that all such linear preservers take the form $\varphi \left(A\right)=\left(Q\oplus \left[1\right]\right)A\left({Q}^{T}\oplus \left[1\right]\right)$, 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