Linear Algebra and its Applications最新文献

{"title":"A connection between the Aα-spectrum and Lovász theta number","authors":"Gabriel Coutinho ,&nbsp;Thiago Oliveira","doi":"10.1016/j.laa.2025.04.007","DOIUrl":"10.1016/j.laa.2025.04.007","url":null,"abstract":"<div><div>We show that the smallest <em>α</em> so that <span><math><mi>α</mi><mi>D</mi><mo>+</mo><mo>(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo>)</mo><mi>A</mi><mo>≽</mo><mn>0</mn></math></span> is at least <span><math><mn>1</mn><mo>/</mo><mi>ϑ</mi><mo>(</mo><mover><mrow><mi>G</mi></mrow><mo>‾</mo></mover><mo>)</mo></math></span>, significantly improving upon a result due to Nikiforov and Rojo (2017). In fact, we display a stronger connection: if the nonzero entries of <em>A</em> are allowed to vary and those of <em>D</em> vary accordingly, then we show that this smallest <em>α</em> is in fact equal to <span><math><mn>1</mn><mo>/</mo><mi>ϑ</mi><mo>(</mo><mover><mrow><mi>G</mi></mrow><mo>‾</mo></mover><mo>)</mo></math></span>. We also show other results obtained as an application of this optimization framework, including a connection to the well-known quadratic formulation for <span><math><mi>ω</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> due to Motzkin and Straus (1964).</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"719 ","pages":"Pages 93-102"},"PeriodicalIF":1.0,"publicationDate":"2025-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143870531","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":"Projections onto polyhedral sets: An improved finite step method and new distributed projection methods","authors":"Yongchao Yu ,&nbsp;Chongyang Wang","doi":"10.1016/j.laa.2025.04.009","DOIUrl":"10.1016/j.laa.2025.04.009","url":null,"abstract":"<div><div>A polyhedral set is the intersection of a finite number of closed half-spaces. It is very difficult to obtain the projection of any point onto a general polyhedral set, especially when the polyhedral set is formed by a large number of closed half-spaces. In this work, we focus on the theoretical aspects of the projection problem itself and of related methods for solving it. The first part of this work is to systematically study various optimality conditions on the projection problem by using the projection theorem. The second part of this work is to design a safe and verifiable screening rule to improve the computational efficiency of Rutkowski's finite step method. In the third part of this work, we introduce a graph-based parameterized operator and prove its conical averagedness. We then propose the convergent scheme of the Krasnosel'skiĭ–Mann fixed point iteration of this operator to find the projection. We also point out that, if we take incidence matrices of graphs as decomposition matrices in the graph-based scheme, the scheme has satisfactory distributability. Several special connected graph networks are provided and under their guidance, new explicit distributed projection methods are shown. These graph-based distributed schemes and methods are also extended to solve the problem of projecting onto finitely generated cones.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"719 ","pages":"Pages 34-65"},"PeriodicalIF":1.0,"publicationDate":"2025-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143870527","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":"Convergence of complementable operators","authors":"Sachin Manjunath Naik,&nbsp;P. Sam Johnson","doi":"10.1016/j.laa.2025.04.008","DOIUrl":"10.1016/j.laa.2025.04.008","url":null,"abstract":"<div><div>Complementable operators extend classical matrix decompositions, such as the Schur complement, to the setting of infinite-dimensional Hilbert spaces, thereby broadening their applicability in various mathematical and physical contexts. This paper focuses on the convergence properties of complementable operators, investigating when the limit of sequence of complementable operators remains complementable. We also explore the convergence of sequences and series of powers of complementable operators, providing new insights into their convergence behavior. Additionally, we examine the conditions under which the set of complementable operators is the subset of set of boundary points of the set of non-complementable operators with respect to the strong operator topology. The paper further explores the topological structure of the subset of complementable operators, offering a characterization of its closed subsets.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"719 ","pages":"Pages 66-92"},"PeriodicalIF":1.0,"publicationDate":"2025-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143870528","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":"Counting core sets in matrix rings over finite fields","authors":"Roswitha Rissner ,&nbsp;Nicholas J. Werner","doi":"10.1016/j.laa.2025.04.006","DOIUrl":"10.1016/j.laa.2025.04.006","url":null,"abstract":"<div><div>Let <em>R</em> be a commutative ring and <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span> be the ring of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices with entries from <em>R</em>. For each <span><math><mi>S</mi><mo>⊆</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span>, we consider its (generalized) null ideal <span><math><mi>N</mi><mo>(</mo><mi>S</mi><mo>)</mo></math></span>, which is the set of all polynomials <em>f</em> with coefficients from <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span> with the property that <span><math><mi>f</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span> for all <span><math><mi>A</mi><mo>∈</mo><mi>S</mi></math></span>. The set <em>S</em> is said to be core if <span><math><mi>N</mi><mo>(</mo><mi>S</mi><mo>)</mo></math></span> is a two-sided ideal of <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo><mo>[</mo><mi>x</mi><mo>]</mo></math></span>. It is not known how common core sets are among all subsets of <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span>. We study this problem for <span><math><mn>2</mn><mo>×</mo><mn>2</mn></math></span> matrices over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>, where <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> is the finite field with <em>q</em> elements. We provide exact counts for the number of core subsets of each similarity class of <span><math><msub><mrow><mi>M</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></math></span>. While not every subset of <span><math><msub><mrow><mi>M</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></math></span> is core, we prove that as <span><math><mi>q</mi><mo>→</mo><mo>∞</mo></math></span>, the probability that a subset of <span><math><msub><mrow><mi>M</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></math></span> is core approaches 1. Thus, asymptotically in <em>q</em>, almost all subsets of <span><math><msub><mrow><mi>M</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></math></span> are core.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"720 ","pages":"Pages 1-25"},"PeriodicalIF":1.0,"publicationDate":"2025-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143869167","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":"Numerical methods for eigenvalues of singular polynomial eigenvalue problems","authors":"Michiel E. Hochstenbach ,&nbsp;Christian Mehl ,&nbsp;Bor Plestenjak","doi":"10.1016/j.laa.2025.04.002","DOIUrl":"10.1016/j.laa.2025.04.002","url":null,"abstract":"<div><div>Recently, three numerical methods for the computation of eigenvalues of singular matrix pencils, based on a rank-completing perturbation, a rank-projection, or an augmentation have been developed. We show that all three approaches can be generalized to treat singular polynomial eigenvalue problems. The common denominator of all three approaches is a transformation of a singular into a regular matrix polynomial whose eigenvalues are a disjoint union of the eigenvalues of the singular polynomial, called true eigenvalues, and additional fake eigenvalues. The true eigenvalues can then be separated from the fake eigenvalues using information on the corresponding left and right eigenvectors. We illustrate the approaches on several interesting applications, including bivariate polynomial systems and ZGV points.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"719 ","pages":"Pages 1-33"},"PeriodicalIF":1.0,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143860059","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":"Spectral w-variation of trees in two places","authors":"Parameswar Basumatary ,&nbsp;Debajit Kalita","doi":"10.1016/j.laa.2025.04.003","DOIUrl":"10.1016/j.laa.2025.04.003","url":null,"abstract":"<div><div>This article introduces the concept of spectral <em>w</em>-variation of weighted graphs. A weighted graph <em>G</em> is said to have spectral <em>w</em>-variation in two places if adding an edge of positive weight <em>w</em> between two nonadjacent vertices of <em>G</em>, or increasing the weight of an existing edge by <em>w</em>, results in an increase of two Laplacian eigenvalues of <em>G</em> equally by <em>w</em> while keeping the other eigenvalues unchanged. This article characterizes the weighted graphs that have spectral <em>w</em>-variation in two places. It is proved that spectral <em>w</em>-variation in two places does not occur by increasing the weight of an existing edge in any weighted tree. Mainly, the article determines the weighted trees that have spectral <em>w</em>-variation in two places. As an application, we supply constructions of few classes of weighted trees with weights from the interval <span><math><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> in which spectral <em>w</em>-variation occurs in two places.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"718 ","pages":"Pages 81-103"},"PeriodicalIF":1.0,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143860337","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":"Periodicity and circulant matrices in the Riordan array of a polynomial","authors":"Nikolai A. Krylov","doi":"10.1016/j.laa.2025.04.005","DOIUrl":"10.1016/j.laa.2025.04.005","url":null,"abstract":"<div><div>We consider Riordan arrays <span><math><mo>(</mo><mn>1</mn><mo>/</mo><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>t</mi></mrow><mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>)</mo><mo>,</mo><mspace></mspace><mi>t</mi><mi>p</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo></math></span>. These are infinite lower triangular matrices determined by the formal power series <span><math><mn>1</mn><mo>/</mo><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>t</mi></mrow><mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> and a polynomial <span><math><mi>p</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> of degree <em>d</em>. Columns of such a matrix are eventually periodic sequences with a period of <span><math><mi>d</mi><mo>+</mo><mn>1</mn></math></span>, and circulant matrices are used to describe the long term behavior of such periodicity when the column's index grows indefinitely. We also discuss some combinatorially interesting sequences that appear through the corresponding A - and Z - sequences of such Riordan arrays.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"718 ","pages":"Pages 58-80"},"PeriodicalIF":1.0,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143834143","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":"Improved bounds for randomized Schatten norm estimation of numerically low-rank matrices","authors":"Ya-Chi Chu,&nbsp;Alice Cortinovis","doi":"10.1016/j.laa.2025.04.001","DOIUrl":"10.1016/j.laa.2025.04.001","url":null,"abstract":"<div><div>In this work, we analyze the variance of a stochastic estimator for computing Schatten norms of matrices. The estimator extracts information from a single sketch of the matrix, that is, the product of the matrix with a few standard Gaussian random vectors. While this estimator has been proposed and used in the literature before, the existing variance bounds are often pessimistic. Our work provides a new upper bound and estimates of the variance of this estimator. These theoretical findings are supported by numerical experiments, demonstrating that the new bounds are significantly tighter than the existing ones in the case of numerically low-rank matrices.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"717 ","pages":"Pages 68-93"},"PeriodicalIF":1.0,"publicationDate":"2025-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143816586","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":"Employing star complements in search for graphs with fixed rank","authors":"Zoran Stanić","doi":"10.1016/j.laa.2025.03.021","DOIUrl":"10.1016/j.laa.2025.03.021","url":null,"abstract":"<div><div>A star complement in a graph <em>G</em> of order <em>n</em> is an induced subgraph <em>H</em> of order <em>t</em>, such that <em>μ</em> is an eigenvalue of multiplicity <span><math><mi>n</mi><mo>−</mo><mi>t</mi></math></span> of <em>G</em>, but not an eigenvalue of <em>H</em>. We use an idea of Torgašev to develop an algorithm based on star complements to characterize graphs with given rank (i.e., the number of non-zero eigenvalues of the adjacency matrix) or given number of eigenvalues distinct from −1. As a demonstration, we re-prove some known results concerning graphs with a comparatively small rank. By the same method, we characterize graphs having at most 6 eigenvalues distinct from −1. Comparisons with existing results are provided.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"718 ","pages":"Pages 14-29"},"PeriodicalIF":1.0,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143817518","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":"Representation of zeros of a copositive matrix via maximal cliques of a graph","authors":"Kostyukova O.I. ,&nbsp;Tchemisova T.V.","doi":"10.1016/j.laa.2025.03.020","DOIUrl":"10.1016/j.laa.2025.03.020","url":null,"abstract":"<div><div>There is a strong connection between copositive matrices and graph theory. Copositive matrices provide a powerful tool for formulating and approximating various challenging graph-related problems. In return, graph theory offers a rich set of concepts and techniques that can be used to explore important properties of copositive matrices, such as their eigenvalues and spectra.</div><div>This paper presents new insights into this interplay. Specifically, we focus on the set of all zeros of a copositive matrix, examining its properties and demonstrating that it can be expressed as a union of convex hulls of certain subsets of minimal zeros. We further show that these subsets are closely linked to the maximal cliques of a special graph, constructed based on the minimal zeros of the matrix.</div><div>An algorithm is explicitly described for constructing the complete set of normalized zeros and minimal zeros of a copositive matrix.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"717 ","pages":"Pages 40-55"},"PeriodicalIF":1.0,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143777425","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}