Pub Date : 2024-10-10DOI: 10.1016/j.laa.2024.10.008
Gabor Lippner, Yujia Shi
We quantify the effect of weighted loops at the source and target nodes of a graph on the strength of quantum state transfer between these vertices. We give lower bounds on loop weights that guarantee strong transfer fidelity that works for any graph where this protocol is feasible. By considering local spectral symmetry, we show that the required weight size depends only on the maximum degree of the graph and, in some less favorable cases, the distance between vertices. Additionally, we explore the duration for which transfer strength remains above a specified threshold.
{"title":"Strong quantum state transfer on graphs via loop edges","authors":"Gabor Lippner, Yujia Shi","doi":"10.1016/j.laa.2024.10.008","DOIUrl":"10.1016/j.laa.2024.10.008","url":null,"abstract":"<div><div>We quantify the effect of weighted loops at the source and target nodes of a graph on the strength of quantum state transfer between these vertices. We give lower bounds on loop weights that guarantee strong transfer fidelity that works for any graph where this protocol is feasible. By considering local spectral symmetry, we show that the required weight size depends only on the maximum degree of the graph and, in some less favorable cases, the distance between vertices. Additionally, we explore the duration for which transfer strength remains above a specified threshold.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"704 ","pages":"Pages 77-91"},"PeriodicalIF":1.0,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142533204","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}
Let F be the finite field of order q and be the set of matrices of rank r over the field F. For and , let In this article, we solve the problem of determining the cardinality of . We also solve the generalization of the problem to rectangular matrices.
{"title":"On the cardinality of matrices with prescribed rank and partial trace over a finite field","authors":"Kumar Balasubramanian , Krishna Kaipa , Himanshi Khurana","doi":"10.1016/j.laa.2024.10.011","DOIUrl":"10.1016/j.laa.2024.10.011","url":null,"abstract":"<div><div>Let <em>F</em> be the finite field of order <em>q</em> and <span><math><mi>M</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> be the set of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices of rank <em>r</em> over the field <em>F</em>. For <span><math><mi>α</mi><mo>∈</mo><mi>F</mi></math></span> and <span><math><mi>A</mi><mo>∈</mo><mi>M</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span>, let<span><span><span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>r</mi></mrow><mrow><mi>α</mi></mrow></msubsup><mo>=</mo><mrow><mo>{</mo><mi>X</mi><mo>∈</mo><mi>M</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>F</mi><mo>)</mo><mo>|</mo><mi>Tr</mi><mo>(</mo><mi>A</mi><mi>X</mi><mo>)</mo><mo>=</mo><mi>α</mi><mo>}</mo></mrow><mo>.</mo></math></span></span></span> In this article, we solve the problem of determining the cardinality of <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>r</mi></mrow><mrow><mi>α</mi></mrow></msubsup></math></span>. We also solve the generalization of the problem to rectangular matrices.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"704 ","pages":"Pages 35-57"},"PeriodicalIF":1.0,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142444534","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}
Pub Date : 2024-10-09DOI: 10.1016/j.laa.2024.10.006
Clément de Seguins Pazzis
Let be a field, and be integers. In a recent article, Rubei has determined, when is the field of real numbers, the greatest possible dimension for an affine subspace of n–by–p matrices with entries in in which all the elements have rank r. In this note, we generalize her result to an arbitrary field with more than elements, and we classify the spaces that reach the maximal dimension as a function of the classification of the affine subspaces of invertible matrices of with dimension . The latter is known to be connected to the classification of nonisotropic quadratic forms over up to congruence.
设 F 是一个域，n≥p≥r>0 是整数。在最近的一篇文章中，鲁贝确定了当 F 是实数域时，所有元素都有秩 r 的 n-by-p 矩阵的仿射子空间的最大可能维度。在本注释中，我们将她的结果推广到元素多于 r+1 的任意域，并将达到最大维度的空间分类为维度为 (s2) 的 Ms(F) 可逆矩阵的仿射子空间分类的函数。众所周知，后者与 F 上非各向同性二次型的分类有关。
{"title":"On affine spaces of rectangular matrices with constant rank","authors":"Clément de Seguins Pazzis","doi":"10.1016/j.laa.2024.10.006","DOIUrl":"10.1016/j.laa.2024.10.006","url":null,"abstract":"<div><div>Let <span><math><mi>F</mi></math></span> be a field, and <span><math><mi>n</mi><mo>≥</mo><mi>p</mi><mo>≥</mo><mi>r</mi><mo>></mo><mn>0</mn></math></span> be integers. In a recent article, Rubei has determined, when <span><math><mi>F</mi></math></span> is the field of real numbers, the greatest possible dimension for an affine subspace of <em>n</em>–by–<em>p</em> matrices with entries in <span><math><mi>F</mi></math></span> in which all the elements have rank <em>r</em>. In this note, we generalize her result to an arbitrary field with more than <span><math><mi>r</mi><mo>+</mo><mn>1</mn></math></span> elements, and we classify the spaces that reach the maximal dimension as a function of the classification of the affine subspaces of invertible matrices of <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math></span> with dimension <span><math><mo>(</mo><mtable><mtr><mtd><mi>s</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></math></span>. The latter is known to be connected to the classification of nonisotropic quadratic forms over <span><math><mi>F</mi></math></span> up to congruence.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"704 ","pages":"Pages 58-76"},"PeriodicalIF":1.0,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142444535","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}
Pub Date : 2024-10-09DOI: 10.1016/j.laa.2024.10.003
Deepak K. D. , Kenta Kojin , Michio Seto
In this short paper, we discuss relation between an invariant distance of the bidisk and Kreĭn space geometry. In particular, an interpolation theorem for rational maps with respect to our invariant distance is proven.
{"title":"A note on an invariant distance of the bidisk","authors":"Deepak K. D. , Kenta Kojin , Michio Seto","doi":"10.1016/j.laa.2024.10.003","DOIUrl":"10.1016/j.laa.2024.10.003","url":null,"abstract":"<div><div>In this short paper, we discuss relation between an invariant distance of the bidisk and Kreĭn space geometry. In particular, an interpolation theorem for rational maps with respect to our invariant distance is proven.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"703 ","pages":"Pages 619-626"},"PeriodicalIF":1.0,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142434517","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}
Pub Date : 2024-10-09DOI: 10.1016/j.laa.2024.10.004
Pascal Koiran
A tuple of matrices of size is said to be a commuting extension of a tuple of matrices of size if the pairwise commute and each sits in the upper left corner of a block decomposition of (here, r and n are two arbitrary integers with ). This notion was discovered and rediscovered in several contexts including algebraic complexity theory (in Strassen's work on tensor rank), in numerical analysis for the construction of cubature formulas and in quantum mechanics for the study of computational methods and the study of the so-called “quantum Zeno dynamics.” Commuting extensions have also attracted the attention of the linear algebra community. In this paper we present 3 types of results:
(i)
Theorems on the uniqueness of commuting extensions for three matrices or more.
(ii)
Algorithms for the computation of commuting extensions of minimal size. These algorithms work under the same assumptions as our uniqueness theorems. They are applicable up to , and are apparently the first provably efficient algorithms for this problem applicable beyond .
(iii)
A genericity theorem showing that our algorithms and uniqueness theorems can be applied to a wide range of input matrices.
大小为 r×r 的矩阵元组 (Z1，...，Zp) 是大小为 n×n 的矩阵元组 (A1，...，Ap) 的换向扩展，如果 Zi 成对换向，并且每个 Ai 都位于 Zi 的块分解的左上角（这里，r 和 n 是两个任意整数，n<r）。这一概念在多个领域被发现和重新发现，包括代数复杂性理论（在斯特拉森关于张量秩的研究中）、用于构造立方公式的数值分析，以及用于研究计算方法和所谓 "量子芝诺动力学 "的量子力学。换元扩展也引起了线性代数界的关注。在本文中，我们提出了三类结果：(i) 三个或更多矩阵的换元扩展唯一性定理；(ii) 计算最小尺寸换元扩展的算法。这些算法的假设条件与我们的唯一性定理相同。(iii)通用性定理表明我们的算法和唯一性定理可以应用于广泛的输入矩阵。
{"title":"On the uniqueness and computation of commuting extensions","authors":"Pascal Koiran","doi":"10.1016/j.laa.2024.10.004","DOIUrl":"10.1016/j.laa.2024.10.004","url":null,"abstract":"<div><div>A tuple <span><math><mo>(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math></span> of matrices of size <span><math><mi>r</mi><mo>×</mo><mi>r</mi></math></span> is said to be a <em>commuting extension</em> of a tuple <span><math><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math></span> of matrices of size <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> if the <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> pairwise commute and each <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> sits in the upper left corner of a block decomposition of <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> (here, <em>r</em> and <em>n</em> are two arbitrary integers with <span><math><mi>n</mi><mo><</mo><mi>r</mi></math></span>). This notion was discovered and rediscovered in several contexts including algebraic complexity theory (in Strassen's work on tensor rank), in numerical analysis for the construction of cubature formulas and in quantum mechanics for the study of computational methods and the study of the so-called “quantum Zeno dynamics.” Commuting extensions have also attracted the attention of the linear algebra community. In this paper we present 3 types of results:<ul><li><span>(i)</span><span><div>Theorems on the uniqueness of commuting extensions for three matrices or more.</div></span></li><li><span>(ii)</span><span><div>Algorithms for the computation of commuting extensions of minimal size. These algorithms work under the same assumptions as our uniqueness theorems. They are applicable up to <span><math><mi>r</mi><mo>=</mo><mn>4</mn><mi>n</mi><mo>/</mo><mn>3</mn></math></span>, and are apparently the first provably efficient algorithms for this problem applicable beyond <span><math><mi>r</mi><mo>=</mo><mi>n</mi><mo>+</mo><mn>1</mn></math></span>.</div></span></li><li><span>(iii)</span><span><div>A genericity theorem showing that our algorithms and uniqueness theorems can be applied to a wide range of input matrices.</div></span></li></ul></div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"703 ","pages":"Pages 645-666"},"PeriodicalIF":1.0,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142434519","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}
Pub Date : 2024-10-09DOI: 10.1016/j.laa.2024.10.007
Tikesh Verma , Debasisha Mishra , Michael Tsatsomeros
Let an complex matrix A be such that is invertible. The Cayley transform of A, denoted by , is defined as Fallat and Tsatsomeros (2002) [5] and Mondal et al. (2024) [15] studied the Cayley transform of a matrix A in the context of P-matrices, H-matrices, M-matrices, totally positive matrices, positive definite matrices, almost skew-Hermitian matrices, and semipositive matrices. In this paper, the investigation of the Cayley transform is continued for Toeplitz matrices, circulant matrices, unipotent matrices, and dual matrices. An expression of the Cayley transform for dual matrices is established. It is shown that the Cayley transform of a dual symmetric matrix is always a dual symmetric matrix. The Cayley transform of a dual skew-symmetric matrix is discussed. The results are illustrated with examples.
{"title":"Cayley transform for Toeplitz and dual matrices","authors":"Tikesh Verma , Debasisha Mishra , Michael Tsatsomeros","doi":"10.1016/j.laa.2024.10.007","DOIUrl":"10.1016/j.laa.2024.10.007","url":null,"abstract":"<div><div>Let an <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> complex matrix <em>A</em> be such that <span><math><mi>I</mi><mo>+</mo><mi>A</mi></math></span> is invertible. The Cayley transform of <em>A</em>, denoted by <span><math><mi>C</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, is defined as<span><span><span><math><mi>C</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><msup><mrow><mo>(</mo><mi>I</mi><mo>+</mo><mi>A</mi><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>I</mi><mo>−</mo><mi>A</mi><mo>)</mo><mo>.</mo></math></span></span></span> Fallat and Tsatsomeros (2002) <span><span>[5]</span></span> and Mondal et al. (2024) <span><span>[15]</span></span> studied the Cayley transform of a matrix <em>A</em> in the context of P-matrices, H-matrices, M-matrices, totally positive matrices, positive definite matrices, almost skew-Hermitian matrices, and semipositive matrices. In this paper, the investigation of the Cayley transform is continued for Toeplitz matrices, circulant matrices, unipotent matrices, and dual matrices. An expression of the Cayley transform for dual matrices is established. It is shown that the Cayley transform of a dual symmetric matrix is always a dual symmetric matrix. The Cayley transform of a dual skew-symmetric matrix is discussed. The results are illustrated with examples.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"703 ","pages":"Pages 627-644"},"PeriodicalIF":1.0,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142434518","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}
Pub Date : 2024-10-09DOI: 10.1016/j.laa.2024.10.005
Ilja Gogić , Tatjana Petek , Mateo Tomašević
Let be the algebra of complex matrices. We consider arbitrary subalgebras of which contain the algebra of all upper-triangular matrices (i.e. block upper-triangular subalgebras), and their Jordan embeddings. We first describe Jordan embeddings as maps of the form or , where is an invertible matrix, and then we obtain a simple criteria of when one block upper-triangular subalgebra Jordan-embeds into another (and in that case we describe the form of such embeddings). As a main result, we characterize Jordan embeddings (when ) as continuous injective maps which preserve commutativity and spectrum. We show by counterexamples that all these assumptions are indispensable (unless when injectivity is superfluous).
{"title":"Characterizing Jordan embeddings between block upper-triangular subalgebras via preserving properties","authors":"Ilja Gogić , Tatjana Petek , Mateo Tomašević","doi":"10.1016/j.laa.2024.10.005","DOIUrl":"10.1016/j.laa.2024.10.005","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> be the algebra of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> complex matrices. We consider arbitrary subalgebras <span><math><mi>A</mi></math></span> of <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> which contain the algebra of all upper-triangular matrices (i.e. block upper-triangular subalgebras), and their Jordan embeddings. We first describe Jordan embeddings <span><math><mi>ϕ</mi><mo>:</mo><mi>A</mi><mo>→</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> as maps of the form <span><math><mi>ϕ</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><mi>T</mi><mi>X</mi><msup><mrow><mi>T</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> or <span><math><mi>ϕ</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><mi>T</mi><msup><mrow><mi>X</mi></mrow><mrow><mi>t</mi></mrow></msup><msup><mrow><mi>T</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span>, where <span><math><mi>T</mi><mo>∈</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is an invertible matrix, and then we obtain a simple criteria of when one block upper-triangular subalgebra Jordan-embeds into another (and in that case we describe the form of such embeddings). As a main result, we characterize Jordan embeddings <span><math><mi>ϕ</mi><mo>:</mo><mi>A</mi><mo>→</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> (when <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span>) as continuous injective maps which preserve commutativity and spectrum. We show by counterexamples that all these assumptions are indispensable (unless <span><math><mi>A</mi><mo>=</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> when injectivity is superfluous).</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"704 ","pages":"Pages 192-217"},"PeriodicalIF":1.0,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142533209","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}
Pub Date : 2024-10-05DOI: 10.1016/j.laa.2024.10.001
Priyanka Joshi , Stephen Kirkland , Helena Šmigoc
The region in the complex plane containing the eigenvalues of all 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 n to be a power of another stochastic matrix.
卡尔佩列维奇于 1951 年描述了复平面上包含所有 n×n 随机矩阵特征值的区域，自此该区域被称为卡尔佩列维奇区域。卡尔佩列维奇区域的边界是称为卡尔佩列维奇弧的弧的联合。我们提供了卡尔佩列维奇弧的完整特征，这些弧是其他一些卡尔佩列维奇弧的幂。此外，我们还找到了与 n 阶 Karpelevič 弧相关的最稀疏随机矩阵是另一个随机矩阵的幂的必要条件和充分条件。
{"title":"Powers of Karpelevič arcs and their sparsest realising matrices","authors":"Priyanka Joshi , Stephen Kirkland , 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":"703 ","pages":"Pages 463-503"},"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}
Pub Date : 2024-10-05DOI: 10.1016/j.laa.2024.10.002
Diogo Diniz , Alex Ramos Borges , Eduardo Fonsêca
We study the number of isomorphism classes of gradings on Lie algebras of block-triangular matrices. Let G be a finite abelian group, for we determine the number of isomorphism classes of elementary G-gradings on the Lie algebra of block-triangular matrices over an algebraically closed field of characteristic zero. We study the asymptotic growth of and as a consequence prove that the determines G up to isomorphism. We also study the asymptotic growth of the number of isomorphism classes of G-gradings on and prove that .
我们研究了块三角矩阵的李代数上等级的同构类数。让 G 是一个有限无性群，对于 m∈Ns ，我们确定了在特征为零的代数闭域上的块三角形矩阵的李代数 UT(m)(-) 上的基本 G 级数的同构类数 E(-)(G,m)。我们研究了 E(-)(G,m)的渐近增长，并由此证明 E(-)(G⋅)决定 G 直到同构。我们还研究了UT(m)(-)上 G-gradings 的同构类数 N(-)(G,m)的渐近增长，并证明了 N(-)(G,m))∼|G|E(-)(G,m)。
{"title":"Counting gradings on Lie algebras of block-triangular matrices","authors":"Diogo Diniz , Alex Ramos Borges , 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":"703 ","pages":"Pages 504-527"},"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}
Pub Date : 2024-10-03DOI: 10.1016/j.laa.2024.09.016
Jeremias Epperlein, Fabian Wirth
We show that the joint spectral radius is pointwise Hölder continuous. In addition, the joint spectral radius is locally Hölder continuous for ε-inflations. In the two-dimensional case, local Hölder continuity holds on the matrix sets with positive joint spectral radius.
{"title":"The joint spectral radius is pointwise Hölder continuous","authors":"Jeremias Epperlein, 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":"704 ","pages":"Pages 92-122"},"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}
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