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

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

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 $x \in \mathbb{R}^3$ such that $x^T(A-λI)x=0$ and both $x$ and $(A-λI)x$ 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 $λ$ 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.

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

Abstract

For a given real matrix , the eigenvalue complementarity problem relative to the Lorentz cone consists of finding a real number and a nonzero vector such that and both and lie in the Lorentz cone, which is comprised of all vectors in forming a or smaller angle with the positive -axis. We refer to the set of all solutions to this eigenvalue complementarity problem as the Lorentz spectrum of . Our work concerns the characterization of the linear preservers of the Lorentz spectrum on the space of real matrices, that is, the linear maps such that the Lorentz spectra of and are the same for all . We have proven that all such linear preservers take the form , where is an orthogonal matrix.
Paper Structure (13 sections, 31 theorems, 181 equations)

This paper contains 13 sections, 31 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. The Lorentz spectrum of a matrix
  3. Matrices in $M_3$ with infinitely many L-eigenvalues
  4. Main results
  5. Image of matrices in $\mathcal{S}_1$ under a linear preserver
  6. Image of matrices in $\mathcal{S}_2$ and $\mathcal{S}_3$ under a linear preserver
  7. Auxiliary results
  8. Image of matrices in $\mathcal{S}_2$ under a linear preserver
  9. Image of matrices in $\mathcal{S}_3$ under a linear preserver
  10. Conclusions
  11. L-spectrum of some special matrices
  12. Results for $n\times n$ matrices
  13. A result for $3\times 3$ matrices

Key Result

Theorem 1

\newlabelthm:bijective1 maribel1 Let $\phi: M_n \rightarrow M_n$ be a linear preserver of the L-spectrum. Then $\phi$ is bijective and $\phi(I_n)= I_n.$

Theorems & Definitions (54)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proof 1
  • Theorem 4
  • Proof 2
  • Theorem 5
  • Theorem 6
  • Lemma 7
  • Proof 3
  • ...and 44 more