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Powers of Karpelevič arcs and their sparsest realising matrices 卡尔佩列维奇弧的幂及其最稀疏实现矩阵
IF 1 3区 数学
Linear Algebra and its Applications Pub Date : 2024-10-05 DOI: 10.1016/j.laa.2024.10.001
Priyanka Joshi , Stephen Kirkland , Helena Šmigoc
{"title":"Powers of Karpelevič arcs and their sparsest realising matrices","authors":"Priyanka Joshi ,&nbsp;Stephen Kirkland ,&nbsp;Helena Šmigoc","doi":"10.1016/j.laa.2024.10.001","DOIUrl":"10.1016/j.laa.2024.10.001","url":null,"abstract":"<div><div>The region in the complex plane containing the eigenvalues of all <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> stochastic matrices was described by Karpelevič in 1951, and it is since then known as the Karpelevič region. The boundary of the Karpelevič region is the union of arcs called the Karpelevič arcs. We provide a complete characterization of the Karpelevič arcs that are powers of some other Karpelevič arc. Furthermore, we find the necessary and sufficient conditions for a sparsest stochastic matrix associated with the Karpelevič arc of order <em>n</em> to be a power of another stochastic matrix.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142424205","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Counting gradings on Lie algebras of block-triangular matrices 块三角矩阵的李代数分级计数
IF 1 3区 数学
Linear Algebra and its Applications Pub Date : 2024-10-05 DOI: 10.1016/j.laa.2024.10.002
Diogo Diniz , Alex Ramos Borges , Eduardo Fonsêca
{"title":"Counting gradings on Lie algebras of block-triangular matrices","authors":"Diogo Diniz ,&nbsp;Alex Ramos Borges ,&nbsp;Eduardo Fonsêca","doi":"10.1016/j.laa.2024.10.002","DOIUrl":"10.1016/j.laa.2024.10.002","url":null,"abstract":"<div><div>We study the number of isomorphism classes of gradings on Lie algebras of block-triangular matrices. Let <em>G</em> be a finite abelian group, for <span><math><mi>m</mi><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>s</mi></mrow></msup></math></span> we determine the number <span><math><msup><mrow><mi>E</mi></mrow><mrow><mo>(</mo><mo>−</mo><mo>)</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>,</mo><mi>m</mi><mo>)</mo></math></span> of isomorphism classes of elementary <em>G</em>-gradings on the Lie algebra <span><math><mi>U</mi><mi>T</mi><msup><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow><mrow><mo>(</mo><mo>−</mo><mo>)</mo></mrow></msup></math></span> of block-triangular matrices over an algebraically closed field of characteristic zero. We study the asymptotic growth of <span><math><msup><mrow><mi>E</mi></mrow><mrow><mo>(</mo><mo>−</mo><mo>)</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>,</mo><mi>m</mi><mo>)</mo></math></span> and as a consequence prove that the <span><math><msup><mrow><mi>E</mi></mrow><mrow><mo>(</mo><mo>−</mo><mo>)</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>,</mo><mo>⋅</mo><mo>)</mo></math></span> determines <em>G</em> up to isomorphism. We also study the asymptotic growth of the number <span><math><msup><mrow><mi>N</mi></mrow><mrow><mo>(</mo><mo>−</mo><mo>)</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>,</mo><mi>m</mi><mo>)</mo></math></span> of isomorphism classes of <em>G</em>-gradings on <span><math><mi>U</mi><mi>T</mi><msup><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow><mrow><mo>(</mo><mo>−</mo><mo>)</mo></mrow></msup></math></span> and prove that <span><math><msup><mrow><mi>N</mi></mrow><mrow><mo>(</mo><mo>−</mo><mo>)</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>,</mo><mi>m</mi><mo>)</mo><mo>)</mo><mo>∼</mo><mo>|</mo><mi>G</mi><mo>|</mo><msup><mrow><mi>E</mi></mrow><mrow><mo>(</mo><mo>−</mo><mo>)</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>,</mo><mi>m</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142424092","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The joint spectral radius is pointwise Hölder continuous 联合光谱半径是点式荷尔德连续的
IF 1 3区 数学
Linear Algebra and its Applications Pub Date : 2024-10-03 DOI: 10.1016/j.laa.2024.09.016
Jeremias Epperlein, Fabian Wirth
{"title":"The joint spectral radius is pointwise Hölder continuous","authors":"Jeremias Epperlein,&nbsp;Fabian Wirth","doi":"10.1016/j.laa.2024.09.016","DOIUrl":"10.1016/j.laa.2024.09.016","url":null,"abstract":"<div><div>We show that the joint spectral radius is pointwise Hölder continuous. In addition, the joint spectral radius is locally Hölder continuous for <em>ε</em>-inflations. In the two-dimensional case, local Hölder continuity holds on the matrix sets with positive joint spectral radius.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142533205","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Universal coacting Hopf algebra of a finite dimensional Lie-Yamaguti algebra 有限维李-山古提代数的普适共作用霍普夫代数
IF 1 3区 数学
Linear Algebra and its Applications Pub Date : 2024-10-03 DOI: 10.1016/j.laa.2024.09.017
Saikat Goswami , Satyendra Kumar Mishra , Goutam Mukherjee
{"title":"Universal coacting Hopf algebra of a finite dimensional Lie-Yamaguti algebra","authors":"Saikat Goswami ,&nbsp;Satyendra Kumar Mishra ,&nbsp;Goutam Mukherjee","doi":"10.1016/j.laa.2024.09.017","DOIUrl":"10.1016/j.laa.2024.09.017","url":null,"abstract":"<div><div>M. E. Sweedler first constructed a universal Hopf algebra of an algebra. It is known that the dual notions to the existing ones play a dominant role in Hopf algebra theory. Yu. I. Manin and D. Tambara introduced the dual notion of Sweedler's construction in separate works. In this paper, we construct a universal algebra for a finite-dimensional Lie-Yamaguti algebra. We demonstrate that this universal algebra possesses a bialgebra structure, leading to a universal coacting Hopf algebra for a finite-dimensional Lie-Yamaguti algebra. Additionally, we develop a representation-theoretic version of our results. As an application, we characterize the automorphism group and classify all abelian group gradings of a finite-dimensional Lie-Yamaguti algebra.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142424013","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Semisimple elements and the little Weyl group of real semisimple Zm-graded Lie algebras 实半简单 Zm 级列的半简单元素和小韦尔群
IF 1 3区 数学
Linear Algebra and its Applications Pub Date : 2024-09-30 DOI: 10.1016/j.laa.2024.09.015
Willem de Graaf , Hông Vân Lê
{"title":"Semisimple elements and the little Weyl group of real semisimple Zm-graded Lie algebras","authors":"Willem de Graaf ,&nbsp;Hông Vân Lê","doi":"10.1016/j.laa.2024.09.015","DOIUrl":"10.1016/j.laa.2024.09.015","url":null,"abstract":"<div><div>We consider the semisimple orbits of a Vinberg <em>θ</em>-representation. First we take the complex numbers as base field. By a case by case analysis we show a technical result stating the equality of two sets of hyperplanes, one corresponding to the restricted roots of a Cartan subspace, the other corresponding to the complex reflections in the (little) Weyl group. The semisimple orbits have representatives in a finite number of sets that correspond to reflection subgroups of the (little) Weyl group. One of the consequences of our technical result is that the elements in a fixed such set all have the same stabilizer in the acting group. Secondly we study what happens when the base field is the real numbers. We look at Cartan subspaces and show that the real Cartan subspaces can be classified by the first Galois cohomology set of the normalizer of a fixed real Cartan subspace. In the real case the orbits can be classified using Galois cohomology. However, in order for that to work we need to know which orbits have a real representative. We show a theorem that characterizes the orbits of homogeneous semisimple elements that do have such a real representative. This closely follows and generalizes a theorem from <span><span>[6]</span></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142424091","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Extreme values of the Fiedler vector on trees 费德勒向量在树木上的极值
IF 1 3区 数学
Linear Algebra and its Applications Pub Date : 2024-09-27 DOI: 10.1016/j.laa.2024.09.014
Roy R. Lederman , Stefan Steinerberger
{"title":"Extreme values of the Fiedler vector on trees","authors":"Roy R. Lederman ,&nbsp;Stefan Steinerberger","doi":"10.1016/j.laa.2024.09.014","DOIUrl":"10.1016/j.laa.2024.09.014","url":null,"abstract":"<div><div>Let <em>G</em> be a tree on <em>n</em> vertices and let <span><math><mi>L</mi><mo>=</mo><mi>D</mi><mo>−</mo><mi>A</mi></math></span> denote the Laplacian matrix on <em>G</em>. The second-smallest eigenvalue <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>&gt;</mo><mn>0</mn></math></span>, also known as the algebraic connectivity, as well as the associated eigenvector have been of substantial interest. We investigate the question of when the maxima and minima of an associated eigenvector are assumed at the endpoints of the longest path in <em>G</em>. Our results also apply to more general graphs that ‘behave globally’ like a tree but can exhibit more complicated local structure. The crucial new ingredient is a reproducing formula for eigenvectors of graphs.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142424093","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Estimations of Karcher mean by Hadamard product 用 Hadamard 乘积估算 Karcher 平均值
IF 1 3区 数学
Linear Algebra and its Applications Pub Date : 2024-09-27 DOI: 10.1016/j.laa.2024.09.013
Masatoshi Fujii , Yuki Seo , Masaru Tominaga
{"title":"Estimations of Karcher mean by Hadamard product","authors":"Masatoshi Fujii ,&nbsp;Yuki Seo ,&nbsp;Masaru Tominaga","doi":"10.1016/j.laa.2024.09.013","DOIUrl":"10.1016/j.laa.2024.09.013","url":null,"abstract":"<div><div>In this paper, we estimate the difference between the Hadamard product and the Karcher mean of <em>n</em> positive invertible operators on the Hilbert space in terms of the Specht ratio and the Kantorovich constant. Also, we improve the obtained inequalities in the case of <span><math><mi>n</mi><mo>=</mo><mn>2</mn></math></span>. Moreover, we give ratio inequalities of the operator power means by the Hadamard product.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142424204","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Isomorphisms between lattices of hyperinvariant subspaces 超不变子空间网格之间的同构关系
IF 1 3区 数学
Linear Algebra and its Applications Pub Date : 2024-09-23 DOI: 10.1016/j.laa.2024.09.012
David Mingueza , M. Eulàlia Montoro , Alicia Roca
{"title":"Isomorphisms between lattices of hyperinvariant subspaces","authors":"David Mingueza ,&nbsp;M. Eulàlia Montoro ,&nbsp;Alicia Roca","doi":"10.1016/j.laa.2024.09.012","DOIUrl":"10.1016/j.laa.2024.09.012","url":null,"abstract":"<div><div>Given two nilpotent endomorphisms, we determine when their lattices of hyperinvariant subspaces are isomorphic. The study of the lattice of hyperinvariant subspaces can be reduced to the nilpotent case when the endomorphism has a Jordan-Chevalley decomposition; for example, it occurs if the underlying field is the field of complex numbers.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142323640","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Locally supported, quasi-interpolatory bases for the approximation of functions on graphs 用于逼近图上函数的局部支持准插值基
IF 1 3区 数学
Linear Algebra and its Applications Pub Date : 2024-09-17 DOI: 10.1016/j.laa.2024.09.011
E. Fuselier , J.P. Ward
{"title":"Locally supported, quasi-interpolatory bases for the approximation of functions on graphs","authors":"E. Fuselier ,&nbsp;J.P. Ward","doi":"10.1016/j.laa.2024.09.011","DOIUrl":"10.1016/j.laa.2024.09.011","url":null,"abstract":"<div><div>Graph-based approximation methods are of growing interest in many areas, including transportation, biological and chemical networks, financial models, image processing, network flows, and more. In these applications, often a basis for the approximation space is not available analytically and must be computed. We propose perturbations of Lagrange bases on graphs, where the Lagrange functions come from a class of functions analogous to classical splines. The basis functions we consider have local support, with each basis function obtained by solving a small energy minimization problem related to a differential operator on the graph. We present <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span> error estimates between the local basis and the corresponding interpolatory Lagrange basis functions in cases where the underlying graph satisfies an assumption on the connections of vertices where the function is not known, and the theoretical bounds are examined further in numerical experiments. Included in our analysis is a mixed-norm inequality for positive definite matrices that is tighter than the general estimate <span><math><msub><mrow><mo>‖</mo><mi>A</mi><mo>‖</mo></mrow><mrow><mo>∞</mo></mrow></msub><mo>≤</mo><msqrt><mrow><mi>n</mi></mrow></msqrt><msub><mrow><mo>‖</mo><mi>A</mi><mo>‖</mo></mrow><mrow><mn>2</mn></mrow></msub></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142319735","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Degradable strong entanglement breaking maps 可降解的强纠缠断裂图
IF 1 3区 数学
Linear Algebra and its Applications Pub Date : 2024-09-17 DOI: 10.1016/j.laa.2024.09.006
Repana Devendra , Gunjan Sapra , K. Sumesh
{"title":"Degradable strong entanglement breaking maps","authors":"Repana Devendra ,&nbsp;Gunjan Sapra ,&nbsp;K. Sumesh","doi":"10.1016/j.laa.2024.09.006","DOIUrl":"10.1016/j.laa.2024.09.006","url":null,"abstract":"<div><p>In this paper, we provide a structure theorem and various characterizations of degradable strong entanglement breaking maps on separable Hilbert spaces. In the finite-dimensional case, we prove that unital degradable entanglement breaking maps are precisely the <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-extreme points of the convex set of unital entanglement breaking maps on matrix algebras. Consequently, we get a structure for unital degradable positive partial transpose (PPT) maps.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142239038","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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