{"title":"单元段特征值之和的凸性","authors":"Gabriel Larotonda, Martin Miglioli","doi":"arxiv-2408.16906","DOIUrl":null,"url":null,"abstract":"For a $n\\times n$ unitary matrix $u=e^z$ with $z$ skew-Hermitian, the angles\nof $u$ are the arguments of its spectrum, i.e. the spectrum of $-iz$. For $1\\le\nm\\le n$, we show that $s_m(t)$, the sum of the first $m$ angles of the path\n$t\\mapsto e^{tx}e^y$ of unitary matrices, is a convex function of $t$ (provided\nthe path stays in a vecinity of the identity matrix). This vecinity is\ndescribed in terms of the opertor norm of matrices, and it is optimal. We show\nthat the when all the maps $t\\mapsto s_m(t)$ are linear, then $x$ commutes with\n$y$. Several application to unitarily invariant norms in the unitary group are\ngiven. Then we extend these applications to $Ad$-invariant Finsler norms in the\nspecial unitary group of matrices. This last result is obtained by proving that\nany $Ad$-invariant Finsler norm in a compact semi-simple Lie group $K$ is the\nsupremum of a family of what we call orbit norms, induced by the Killing form\nof $K$.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Convexity of sums of eigenvalues of a segment of unitaries\",\"authors\":\"Gabriel Larotonda, Martin Miglioli\",\"doi\":\"arxiv-2408.16906\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For a $n\\\\times n$ unitary matrix $u=e^z$ with $z$ skew-Hermitian, the angles\\nof $u$ are the arguments of its spectrum, i.e. the spectrum of $-iz$. For $1\\\\le\\nm\\\\le n$, we show that $s_m(t)$, the sum of the first $m$ angles of the path\\n$t\\\\mapsto e^{tx}e^y$ of unitary matrices, is a convex function of $t$ (provided\\nthe path stays in a vecinity of the identity matrix). This vecinity is\\ndescribed in terms of the opertor norm of matrices, and it is optimal. We show\\nthat the when all the maps $t\\\\mapsto s_m(t)$ are linear, then $x$ commutes with\\n$y$. Several application to unitarily invariant norms in the unitary group are\\ngiven. Then we extend these applications to $Ad$-invariant Finsler norms in the\\nspecial unitary group of matrices. This last result is obtained by proving that\\nany $Ad$-invariant Finsler norm in a compact semi-simple Lie group $K$ is the\\nsupremum of a family of what we call orbit norms, induced by the Killing form\\nof $K$.\",\"PeriodicalId\":501444,\"journal\":{\"name\":\"arXiv - MATH - Metric Geometry\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Metric Geometry\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2408.16906\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Metric Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.16906","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Convexity of sums of eigenvalues of a segment of unitaries
For a $n\times n$ unitary matrix $u=e^z$ with $z$ skew-Hermitian, the angles
of $u$ are the arguments of its spectrum, i.e. the spectrum of $-iz$. For $1\le
m\le n$, we show that $s_m(t)$, the sum of the first $m$ angles of the path
$t\mapsto e^{tx}e^y$ of unitary matrices, is a convex function of $t$ (provided
the path stays in a vecinity of the identity matrix). This vecinity is
described in terms of the opertor norm of matrices, and it is optimal. We show
that the when all the maps $t\mapsto s_m(t)$ are linear, then $x$ commutes with
$y$. Several application to unitarily invariant norms in the unitary group are
given. Then we extend these applications to $Ad$-invariant Finsler norms in the
special unitary group of matrices. This last result is obtained by proving that
any $Ad$-invariant Finsler norm in a compact semi-simple Lie group $K$ is the
supremum of a family of what we call orbit norms, induced by the Killing form
of $K$.