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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2209.00214 |
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| _version_ | 1866911801752420352 |
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| author | Bueno, M. I. Faktor, Ben Kommerell, Rhea Li, Runze Veltri, Joey |
| author_facet | Bueno, M. I. Faktor, Ben Kommerell, Rhea Li, Runze Veltri, Joey |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_00214 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Linear maps preserving the Lorentz spectrum of $3 \times 3$ matrices Bueno, M. I. Faktor, Ben Kommerell, Rhea Li, Runze Veltri, Joey Rings and Algebras 15A18, 58C40 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. |
| title | Linear maps preserving the Lorentz spectrum of $3 \times 3$ matrices |
| topic | Rings and Algebras 15A18, 58C40 |
| url | https://arxiv.org/abs/2209.00214 |