保留洛伦兹谱的线性映射:$2\times2$情形

IF 0.7 4区 数学 Q2 Mathematics Electronic Journal of Linear Algebra Pub Date : 2021-11-13 DOI:10.13001/ela.2022.6925
M. Bueno, S. Furtado, Aelita Klausmeier, Joey Veltri
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引用次数: 2

摘要

本文给出了当$n=2$, $W_{n}$为$n\times n$实矩阵的空间$M_{n}$或由对称矩阵构成的$M_{n}$的子空间$S_{n}$时保持洛伦兹谱的线性映射$\phi:W_{n}\rightarrow W_{n}$的完整描述。在这两种情况下,已经证明$\phi(A)=PAP^{-1}$适用于所有$A\in W_{2}$,其中$P$是具有一定结构的矩阵。研究还表明,这些守恒子不会改变洛伦兹特征值的性质(也就是说,它们与洛伦兹锥内部或边界上的洛伦兹特征向量相关联的事实)。这些结果延伸到$n=2$,由Bueno, Furtado和Sivakumar(2021)获得的$n\geq 3$的结果。与情况$n\geq3,$相比,情况$n=2$有一些特殊性,因为$\mathbb{R}^{2}$中的洛伦兹锥是聚体的,与$\mathbb{R}^{n}$中包含的情况相反,与$n\geq3.$中包含的情况相反,因此,$W_n = M_n$上的洛伦兹谱保存器的研究也遵循$M_n$上已知的帕累托谱保存器的描述。
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Linear maps preserving the Lorentz spectrum: the $2 \times 2$ case
In this paper, a complete description of the linear maps $\phi:W_{n}\rightarrow W_{n}$ that preserve the Lorentz spectrum is given when $n=2$, and $W_{n}$ is the space $M_{n}$ of $n\times n$ real matrices or the subspace $S_{n}$ of $M_{n}$ formed by the symmetric matrices. In both cases, it has been shown that $\phi(A)=PAP^{-1}$ for all $A\in W_{2}$, where $P$ is a matrix with a certain structure. It was also shown that such preservers do not change the nature of the Lorentz eigenvalues (that is, the fact that they are associated with Lorentz eigenvectors in the interior or on the boundary of the Lorentz cone). These results extend to $n=2$ those for $n\geq 3$ obtained by Bueno, Furtado, and Sivakumar (2021). The case $n=2$ has some specificities, when compared to the case $n\geq3,$ due to the fact that the Lorentz cone in $\mathbb{R}^{2}$ is polyedral, contrary to what happens when it is contained in $\mathbb{R}^{n}$ with $n\geq3.$ Thus, the study of the Lorentz spectrum preservers on $W_n = M_n$ also follows from the known description of the Pareto spectrum preservers on $M_n$.
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来源期刊
CiteScore
1.20
自引率
14.30%
发文量
45
审稿时长
6-12 weeks
期刊介绍: The journal is essentially unlimited by size. Therefore, we have no restrictions on length of articles. Articles are submitted electronically. Refereeing of articles is conventional and of high standards. Posting of articles is immediate following acceptance, processing and final production approval.
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