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

TL;DR

This paper characterizes the linear transformations on 3x3 matrices that preserve the Lorentz spectrum, revealing they are essentially orthogonal conjugations extended with identity.

Contribution

It provides a complete description of linear spectrum preservers for the Lorentz spectrum on 3x3 matrices, a novel spectral invariance result.

Findings

All spectrum preservers are orthogonal conjugations extended with identity.

The Lorentz spectrum is invariant under specific linear transformations.

The characterization applies to the space of 3x3 real matrices.

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 $\lambda$ and a nonzero vector $x \in \mathbb{R}^3$ such that $x^T(A-\lambda I)x=0$ and both $x$ and $(A-\lambda I)x$ lie in the Lorentz cone, which is comprised of all vectors in $\mathbb{R}^3$ forming a $45^\circ$ 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 $3 \times 3$ real matrices, that is, the linear maps $\phi: M_3 \to M_3$ such that the Lorentz spectra of $A$ and $\phi(A)$ are the same for all $A$. We have proven that all such linear preservers take the form $\phi(A) = (Q \oplus [1])A(Q^T \oplus [1])$, where $Q$ is an orthogonal $2 \times 2$ matrix.