M. Bueno, S. Furtado, Aelita Klausmeier, Joey Veltri
{"title":"保留洛伦兹谱的线性映射:$2\\times2$情形","authors":"M. Bueno, S. Furtado, Aelita Klausmeier, Joey Veltri","doi":"10.13001/ela.2022.6925","DOIUrl":null,"url":null,"abstract":"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 $M_n$.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2021-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"Linear maps preserving the Lorentz spectrum: the $2 \\\\times 2$ case\",\"authors\":\"M. Bueno, S. Furtado, Aelita Klausmeier, Joey Veltri\",\"doi\":\"10.13001/ela.2022.6925\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"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 $M_n$.\",\"PeriodicalId\":50540,\"journal\":{\"name\":\"Electronic Journal of Linear Algebra\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2021-11-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Electronic Journal of Linear Algebra\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.13001/ela.2022.6925\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Electronic Journal of Linear Algebra","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.13001/ela.2022.6925","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"Mathematics","Score":null,"Total":0}
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 $M_n$.
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.