Linear Algebra and its Applications最新文献

{"title":"Linear bijective maps preserving invertibility on pairs of similar matrices","authors":"Constantin Costara","doi":"10.1016/j.laa.2024.12.021","DOIUrl":"10.1016/j.laa.2024.12.021","url":null,"abstract":"<div><div>Let <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span> be a natural number, and denote by <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> the space of all <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices over the complex field. In this paper, we characterize linear bijective maps <em>φ</em> on <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> having the property that if <span><math><mi>A</mi><mo>,</mo><mi>B</mi><mo>∈</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> are similar matrices and <span><math><mi>φ</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> is invertible, then <span><math><mi>φ</mi><mo>(</mo><mi>B</mi><mo>)</mo></math></span> is invertible as well.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 585-593"},"PeriodicalIF":1.0,"publicationDate":"2024-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143165108","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":"Asymptotic spectral properties and preconditioning of an approximated nonlocal Helmholtz equation with fractional Laplacian and variable coefficient wave number μ","authors":"Andrea Adriani ,&nbsp;Rosita L. Sormani ,&nbsp;Cristina Tablino-Possio ,&nbsp;Rolf Krause ,&nbsp;Stefano Serra-Capizzano","doi":"10.1016/j.laa.2024.12.015","DOIUrl":"10.1016/j.laa.2024.12.015","url":null,"abstract":"<div><div>The current study investigates the asymptotic spectral properties of a finite difference approximation of nonlocal Helmholtz equations with a fractional Laplacian and a variable coefficient wave number <em>μ</em>, as it occurs when considering a wave propagation in complex media, characterized by nonlocal interactions and spatially varying wave speeds. More specifically, by using tools from Toeplitz and generalized locally Toeplitz theory, the present research delves into the spectral analysis of nonpreconditioned and preconditioned matrix sequences, with the main novelty regarding a complete picture of the case where <span><math><mi>μ</mi><mo>=</mo><mi>μ</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> is nonconstant. We report numerical evidence supporting the theoretical findings. Finally, open problems and potential extensions in various directions are presented and briefly discussed.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 551-584"},"PeriodicalIF":1.0,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143165107","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}

{"title":"Some stability results for spectral extremal problems of graphs with bounded matching number","authors":"Shixia Jiang ,&nbsp;Xiying Yuan ,&nbsp;Yanni Zhai","doi":"10.1016/j.laa.2024.12.018","DOIUrl":"10.1016/j.laa.2024.12.018","url":null,"abstract":"<div><div>For a set of graphs <span><math><mi>H</mi></math></span>, a graph is called <span><math><mi>H</mi></math></span>-free if it does not contain any member of <span><math><mi>H</mi></math></span> as a subgraph. The maximum value of spectral radius among all <span><math><mi>H</mi></math></span>-free graphs of order <em>n</em> is denoted by <span><math><mrow><mi>spex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span>, and the set of corresponding extremal graphs is denoted by <span><math><mrow><mi>SPEX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span>. In this paper, we give a stability result for graphs in <span><math><mrow><mi>SPEX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> when <span><math><mrow><mi>spex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo><mo>≥</mo><msqrt><mrow><mi>s</mi><mo>(</mo><mi>n</mi><mo>−</mo><mi>s</mi><mo>)</mo></mrow></msqrt></math></span> and <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo><mo>≤</mo><mi>s</mi><mi>n</mi></math></span>. As an application, we may give some characterizations for the graphs in <span><math><mrow><mi>SPEX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mo>{</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>H</mi><mo>}</mo><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> is a matching with <span><math><mi>s</mi><mo>+</mo><mn>1</mn></math></span> edges and <em>H</em> is any non-bipartite graph.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 513-524"},"PeriodicalIF":1.0,"publicationDate":"2024-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143165105","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":"Remarks on generating families of matrix algebras of small orders","authors":"Yaroslav Shitov","doi":"10.1016/j.laa.2024.12.013","DOIUrl":"10.1016/j.laa.2024.12.013","url":null,"abstract":"<div><div>Let <span><math><mi>n</mi><mo>⩽</mo><mn>7</mn></math></span>, and let <em>S</em> be a family of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices over a field <span><math><mi>F</mi></math></span>. I prove that the <span><math><mi>F</mi></math></span>-linear span of<span><span><span><math><msup><mrow><mo>(</mo><mi>S</mi><mo>∪</mo><mo>{</mo><mrow><mi>Id</mi></mrow><mo>}</mo><mo>)</mo></mrow><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msup></math></span></span></span> is the algebra generated by <em>S</em>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 458-462"},"PeriodicalIF":1.0,"publicationDate":"2024-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143165101","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":"Fractional perfect matching and distance spectral radius in graphs","authors":"Lei Zhang ,&nbsp;Yaoping Hou ,&nbsp;Haizhen Ren","doi":"10.1016/j.laa.2024.12.016","DOIUrl":"10.1016/j.laa.2024.12.016","url":null,"abstract":"<div><div>A fractional matching of a graph <em>G</em> is a function <em>f</em> giving each edge a number in <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span> so that <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>e</mi><mo>∈</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo></mrow></msub><mi>f</mi><mo>(</mo><mi>e</mi><mo>)</mo><mo>≤</mo><mn>1</mn></math></span> for each <span><math><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo></math></span> is the set of edges incident to <em>v</em>. In this paper, we give a distance spectral radius condition to guarantee the existence of a fractional perfect matching. This result generalize the result of Lin and Zhang (2021) <span><span>[22]</span></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 480-488"},"PeriodicalIF":1.0,"publicationDate":"2024-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143165103","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":"On the necessary and sufficient conditions for Hadamard-Fischer-Koteljanskii type inequalities","authors":"Phillip Braun ,&nbsp;Hristo Sendov","doi":"10.1016/j.laa.2024.12.017","DOIUrl":"10.1016/j.laa.2024.12.017","url":null,"abstract":"<div><div>This work explores the ratios of products of determinants of principal submatrices of positive definite matrices. We investigate conditions under which these ratios are bounded, particularly revisiting the necessary/sufficient conditions proposed by Johnson and Barrett. This analysis extends to set-theoretic consequences and unboundedness of certain ratios. We also demonstrate how these conditions can be used to prove the boundedness of several known determinantal inequalities. Additionally, we address the optimization problem of finding the supremum of such ratios over all positive definite matrices, formulating it as a linear optimization program. Finally, for completeness, we include the proofs of theorems that appear to have been previously known but lack accessible proofs.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 525-550"},"PeriodicalIF":1.0,"publicationDate":"2024-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143165106","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":"On the local dimensions of solutions of Brent equations","authors":"Xin Li ,&nbsp;Yixin Bao ,&nbsp;Liping Zhang","doi":"10.1016/j.laa.2024.12.011","DOIUrl":"10.1016/j.laa.2024.12.011","url":null,"abstract":"<div><div>Let <span><math><mo>〈</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>〉</mo></math></span> be the matrix multiplication tensor. The solution set of Brent equations corresponds to the tensor decompositions of <span><math><mo>〈</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>〉</mo></math></span>. We study the local dimensions of solutions of the Brent equations over the field of complex numbers. The rank of Jacobian matrix of Brent equations provides an upper bound of the local dimension, which is well-known. We calculate the ranks for some typical known solutions, which are provided in the databases <span><span>[16]</span></span> and <span><span>[17]</span></span>. We show that the automorphism group of the natural algorithm computing <span><math><mo>〈</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>〉</mo></math></span> is <span><math><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>×</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>×</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo><mo>⋊</mo><mi>Q</mi><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span>, <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> are groups of generalized permutation matrices, <span><math><mi>Q</mi><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span> is a subgroup of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> depending on <em>m</em>, <em>n</em> and <em>p</em>. For other algorithms computing <span><math><mo>〈</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>〉</mo></math></span>, some conditions are given, which imply the corresponding automorphism groups are isomorphic to subgroups of <span><math><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>×</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>×</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo><mo>⋊</mo><mi>Q</mi><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span>. So under these conditions, <span><math><msup><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mi>m</mi><mo>−</mo><mi>n</mi><mo>−</mo><mi>p</mi><mo>−</mo><mn>3</mn></math></span> is a lower bound for the local dimensions of solutions of Brent equations. Moreover, the gap between the lower and upper bounds is discussed.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 489-512"},"PeriodicalIF":1.0,"publicationDate":"2024-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143165104","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":"On the maximal ranks of some third-order quaternion tensors","authors":"Y.G. Liang,&nbsp;Yang Zhang","doi":"10.1016/j.laa.2024.12.010","DOIUrl":"10.1016/j.laa.2024.12.010","url":null,"abstract":"<div><div>A complex tensor <em>T</em> can be considered as a quaternion tensor. Consequently, decomposing <em>T</em> using simple quaternion tensors, rather than simple complex tensors, can potentially result in decompositions with a smaller rank. In this paper, we first present an example demonstrating this. Furthermore, we show that the maximal rank of a 3 × 3 × 3 quaternion tensor is 5, and in doing so provide explicit decompositions into simple tensors with several cases. Finally, we provide the maximal ranks for all <span><math><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>×</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>×</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> quaternion tensors with <span><math><mn>2</mn><mo>≤</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>≤</mo><mn>3</mn></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 405-428"},"PeriodicalIF":1.0,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143165099","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":"Eigenvalues of parametric rank-one perturbations of matrix pencils","authors":"Hannes Gernandt ,&nbsp;Carsten Trunk","doi":"10.1016/j.laa.2024.12.012","DOIUrl":"10.1016/j.laa.2024.12.012","url":null,"abstract":"<div><div>We study the behavior of eigenvalues of regular matrix pencils under rank-one perturbations which depend on a scalar parameter. In particular, the change of the algebraic multiplicities, the change of the eigenvalues for small parameter variations, as well as the asymptotic eigenvalue behavior as the parameter tends to infinity, is described. Besides that, an interlacing result for rank-one perturbations of matrix pencils is obtained. Finally, we show how to use these results in the redesign of electrical circuits, like for the low pass filter or for a two-stage CMOS operational amplifier.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 429-457"},"PeriodicalIF":1.0,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143165100","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}

{"title":"Extensions of Riccati diagonal stability","authors":"Ali Algefary ,&nbsp;Jianhong Xu","doi":"10.1016/j.laa.2024.12.014","DOIUrl":"10.1016/j.laa.2024.12.014","url":null,"abstract":"<div><div>As generalizations of Riccati diagonal stability on a matrix pair, the notions of Riccati <em>α</em>-scalar stability and <em>α</em>-diagonal stability are introduced and fully characterized. Further extensions involving different block diagonal structures, simultaneous <em>α</em>-scalar stability, and simultaneous <em>α</em>-diagonal stability are also presented.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 463-479"},"PeriodicalIF":1.0,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143165102","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}