Linear Algebra and its Applications最新文献

{"title":"Limit points of Aα-matrices of graphs","authors":"Elismar R. Oliveira,&nbsp;Vilmar Trevisan","doi":"10.1016/j.laa.2025.08.014","DOIUrl":"10.1016/j.laa.2025.08.014","url":null,"abstract":"<div><div>We study limit points of the spectral radii of <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>-matrices of graphs. Adapting a method used by J. B. Shearer in 1989, we prove a density property of <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>-limit points of caterpillars for <em>α</em> close to zero. Precisely, we show that for <span><math><mi>α</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></math></span> there exists a positive number <span><math><msub><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>α</mi><mo>)</mo><mo>&gt;</mo><mn>2</mn></math></span> such that any value <span><math><mi>λ</mi><mo>&gt;</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>α</mi><mo>)</mo></math></span> is an <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>-limit point. We also determine other intervals whose numbers are all <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>-limit points.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"728 ","pages":"Pages 1-25"},"PeriodicalIF":1.1,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144903259","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}

{"title":"Tridiagonal generalized inverses of singular matrices possessing the triangle property","authors":"A.M. Encinas ,&nbsp;K. Kranthi Priya ,&nbsp;K.C. Sivakumar","doi":"10.1016/j.laa.2025.08.011","DOIUrl":"10.1016/j.laa.2025.08.011","url":null,"abstract":"<div><div>It is known that an invertible real square matrix has the triangle property if and only if the inverse is a tridiagonal matrix. This result has an implicit importance due to the fact that nonsingular tridiagonal matrices arise in a variety of problems in pure and applied mathematics and for this reason they have been extensively studied in the literature. However, the singular case has received comparatively much lesser attention. In particular, there has been little focus on the generalized inverses of such matrices. In this paper, we provide a complete description of those singular matrices possessing the triangle property to have the tridiagonal Moore-Penrose inverse or group inverse. The converse statements are also completely answered.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"727 ","pages":"Pages 268-307"},"PeriodicalIF":1.1,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144904383","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}

{"title":"Construction of exceptional copositive matrices","authors":"Tea Štrekelj ,&nbsp;Aljaž Zalar","doi":"10.1016/j.laa.2025.08.010","DOIUrl":"10.1016/j.laa.2025.08.010","url":null,"abstract":"<div><div>An <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> symmetric matrix <em>A</em> is copositive if the quadratic form <span><math><msup><mrow><mi>x</mi></mrow><mrow><mi>T</mi></mrow></msup><mi>A</mi><mi>x</mi></math></span> is nonnegative on the nonnegative orthant <span><math><msubsup><mrow><mi>R</mi></mrow><mrow><mo>≥</mo><mn>0</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>. The cone of copositive matrices contains the cone of matrices which are the sum of a positive semidefinite matrix and a nonnegative one and the latter contains the cone of completely positive matrices. These are the matrices of the form <span><math><mi>B</mi><msup><mrow><mi>B</mi></mrow><mrow><mi>T</mi></mrow></msup></math></span> for some <span><math><mi>n</mi><mo>×</mo><mi>r</mi></math></span> matrix <em>B</em> with nonnegative entries. The above inclusions are strict for <span><math><mi>n</mi><mo>≥</mo><mn>5</mn></math></span>. The first main result of this article is a free probability inspired construction of exceptional copositive matrices of all sizes ≥5, i.e., copositive matrices that are not the sum of a positive semidefinite matrix and a nonnegative one. The second contribution of this paper addresses the asymptotic ratio of the volume radii of compact sections of the cones of copositive and completely positive matrices. In a previous work by Klep and the authors, it was shown that, by identifying symmetric matrices naturally with quartic even forms, and equipping them with the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> inner product and the Lebesgue measure, the ratio of the volume radii of sections with a suitably chosen hyperplane is bounded below by a constant independent of <em>n</em> as <em>n</em> tends to infinity. In this paper, we complement this result by establishing an analogous bound when the sections of the cones are unit balls in the Frobenius inner product.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"727 ","pages":"Pages 368-384"},"PeriodicalIF":1.1,"publicationDate":"2025-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144904284","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}

{"title":"The Lawrence–Krammer representation is a quantization of the symmetric square of the Burau representation","authors":"Alexandr V. Kosyak","doi":"10.1016/j.laa.2025.08.009","DOIUrl":"10.1016/j.laa.2025.08.009","url":null,"abstract":"<div><div>We show that the Lawrence–Krammer representation is a quantization of the symmetric square of the Burau representation. Here, by quantization we mean changing the natural numbers by <em>q</em>-natural numbers and the corresponding quantization of the Pascal triangle, appearing naturally in representations of the braid groups. This connection allows us to construct new representations of the braid groups.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"727 ","pages":"Pages 203-233"},"PeriodicalIF":1.1,"publicationDate":"2025-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144894874","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}

{"title":"Graded representations of current Lie superalgebra sl(1|2)[t]","authors":"Shushma Rani ,&nbsp;Divya Setia","doi":"10.1016/j.laa.2025.08.007","DOIUrl":"10.1016/j.laa.2025.08.007","url":null,"abstract":"<div><div>In this paper, we study finite-dimensional graded representations of the current Lie superalgebra <span><math><mrow><mi>sl</mi></mrow><mo>(</mo><mn>1</mn><mo>|</mo><mn>2</mn><mo>)</mo><mo>[</mo><mi>t</mi><mo>]</mo></math></span>. We define the notion of super POPs, a combinatorial tool to provide another parametrization of the basis of the local Weyl module given by Brito, Calixto and Macedo. We derive the graded character formula of the local Weyl module for <span><math><mrow><mi>sl</mi></mrow><mo>(</mo><mn>1</mn><mo>|</mo><mn>2</mn><mo>)</mo><mo>[</mo><mi>t</mi><mo>]</mo></math></span>. Furthermore, we construct a short exact sequence of Chari-Venkatesh modules for <span><math><mrow><mi>sl</mi></mrow><mo>(</mo><mn>1</mn><mo>|</mo><mn>2</mn><mo>)</mo><mo>[</mo><mi>t</mi><mo>]</mo></math></span>. As a consequence, we prove that Chari-Venkatesh modules are isomorphic to the fusion product of generalized Kac modules.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"727 ","pages":"Pages 178-202"},"PeriodicalIF":1.1,"publicationDate":"2025-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144885470","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}

{"title":"Metric Frobenius norms and inner products of matrices and linear maps","authors":"Roland Herzog ,&nbsp;Frederik Köhne ,&nbsp;Leonie Kreis ,&nbsp;Anton Schiela","doi":"10.1016/j.laa.2025.08.005","DOIUrl":"10.1016/j.laa.2025.08.005","url":null,"abstract":"<div><div>The Frobenius norm is a frequent choice of norm for matrices. We provide a broader view on the Frobenius norm and Frobenius inner product for linear maps or matrices, and establish their dependence on inner products in the domain and co-domain spaces. These new concepts are termed the metric Frobenius norm and metric Frobenius inner product. We demonstrate that the classical Frobenius norm is merely one particular element of the family of metric Frobenius norms. We also show that the metric Frobenius norm has an interpretation similar to an operator norm of a linear map. While the usual operator norm is defined as the maximal norm response of the map w.r.t. inputs in the unit sphere, the Frobenius norm turns out to measure the average norm response.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"727 ","pages":"Pages 112-128"},"PeriodicalIF":1.1,"publicationDate":"2025-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144828825","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}

{"title":"Homomorphisms from the Coxeter graph","authors":"Marko Orel ,&nbsp;Draženka Višnjić","doi":"10.1016/j.laa.2025.08.003","DOIUrl":"10.1016/j.laa.2025.08.003","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> be the set of all <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> symmetric matrices with coefficients in the binary field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>, and let <span><math><mi>S</mi><mi>G</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> be its subset formed by invertible matrices. Let <span><math><msub><mrow><mover><mrow><mi>Γ</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>n</mi></mrow></msub></math></span> be the graph with the vertex set <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> where a pair of vertices <span><math><mo>{</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>}</mo></math></span> form an edge if and only if <span><math><mrow><mi>rank</mi></mrow><mo>(</mo><mi>A</mi><mo>−</mo><mi>B</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span>. Similarly, let <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> be the subgraph in <span><math><msub><mrow><mover><mrow><mi>Γ</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>n</mi></mrow></msub></math></span>, which is induced by the set <span><math><mi>S</mi><mi>G</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>. Graph <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> generalizes the well-known Coxeter graph, which is isomorphic to <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>. Motivated by research topics in coding theory, matrix theory, and graph theory, this paper represents the first step towards the characterization of all graph homomorphisms <span><math><mi>Φ</mi><mo>:</mo><msub><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>→</mo><msub><mrow><mover><mrow><mi>Γ</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>m</mi></mrow></msub></math></span> where <span><math><mi>n</mi><mo>,</mo><mi>m</mi></math></span> are positive integers. Here, the case <span><math><mi>n</mi><mo>=</mo><mn>3</mn></math></span> is solved.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"727 ","pages":"Pages 129-162"},"PeriodicalIF":1.1,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144860768","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}

{"title":"A hidden variable resultant method for the polynomial multiparameter eigenvalue problem","authors":"Emil Graf,&nbsp;Alex Townsend","doi":"10.1016/j.laa.2025.07.031","DOIUrl":"10.1016/j.laa.2025.07.031","url":null,"abstract":"<div><div>We present a novel, global algorithm for solving polynomial multiparameter eigenvalue problems (PMEPs) by leveraging a hidden variable tensor Dixon resultant framework. Our method transforms a PMEP into one or more univariate polynomial eigenvalue problems, which are solved as generalized eigenvalue problems. Our general approach avoids the need for custom linearizations of PMEPs. We provide rigorous theoretical guarantees for generic PMEPs and give practical strategies for nongeneric systems. Benchmarking on applications from aeroelastic flutter and leaky wave propagation confirms that our algorithm attains high accuracy and robustness while being broadly applicable to many PMEPs.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"727 ","pages":"Pages 37-60"},"PeriodicalIF":1.1,"publicationDate":"2025-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144810385","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}

{"title":"Extremal properties and bounds for the generalized algebraic connectivity of graphs in Euclidean spaces","authors":"J. Ignacio Alvarez-Hamelin ,&nbsp;Juan I. Giribet ,&nbsp;Ignacio Mas ,&nbsp;J. Francisco Presenza","doi":"10.1016/j.laa.2025.07.028","DOIUrl":"10.1016/j.laa.2025.07.028","url":null,"abstract":"<div><div>This article contributes to the study of graph rigidity and its interplay with fundamental graph invariants. Recently, a quantitative measure of graph rigidity in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, termed the generalized algebraic connectivity, was introduced. This development extends the notion of algebraic connectivity—the second-smallest eigenvalue of the Laplacian matrix. In this work, we show that the generalized algebraic connectivity is bounded above by the algebraic connectivity. To capture this relationship, we introduce the <em>d</em>-rigidity ratio, a normalized metric of a graph's rigidity relative to its connectivity. We also investigate the relationship between rigidity and the diameter. In this context, we provide the maximal diameter achievable by rigid graphs and show that generalized path graphs serve as extremal examples. Moreover, we establish a new upper bound for the algebraic connectivity that depends inversely on the diameter and the vertex connectivity. Finally, we derive an upper bound for the algebraic connectivity of generalized path graphs that asymptotically improves existing ones by a factor of four.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"727 ","pages":"Pages 24-36"},"PeriodicalIF":1.1,"publicationDate":"2025-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144810384","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}

{"title":"Symmetries of hypergraphs and some invariant subspaces of matrices associated with hypergraphs","authors":"Anirban Banerjee ,&nbsp;Samiron Parui","doi":"10.1016/j.laa.2025.07.030","DOIUrl":"10.1016/j.laa.2025.07.030","url":null,"abstract":"<div><div>Here, the structural symmetries of a hypergraph are represented through equivalence relations on the vertex set of the hypergraph. A matrix associated with the hypergraph may not reflect a specific structural symmetry. In the context of a given symmetry within a hypergraph, we investigate a collection of matrices that encapsulate information about the symmetry. Our investigation reveals that certain structural symmetries in a hypergraph manifest observable effects on the eigenvalues and eigenvectors of designated matrices associated with the hypergraph. We identify specific matrices where the invariance is a consequence of symmetries present in the hypergraph. These invariant subspaces elucidate analogous behaviours observed in certain clusters of vertices during random walks and other dynamical processes on the hypergraph.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"726 ","pages":"Pages 328-358"},"PeriodicalIF":1.1,"publicationDate":"2025-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144773125","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}