Linear Algebra and its Applications最新文献

{"title":"Almost commuting self-adjoint operators and iterated commutator estimates","authors":"Jakob Geisler","doi":"10.1016/j.laa.2025.09.004","DOIUrl":"10.1016/j.laa.2025.09.004","url":null,"abstract":"<div><div>Given two almost commuting self-adjoint operators, a new method for finding exactly commuting operators is presented. For this, a differential equation for self-adjoint Hilbert-Schmidt operators is introduced. Quantitative results are proven that the exactly commuting operators are close to the old ones in the Hilbert-Schmidt norm. The proof relies on a novel estimate in which the norm of the commutator is bounded from above by the norm of the iterated commutators times a constant. This inequality is proven in finite dimensions and lower bounds for the optimal constants are given.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"727 ","pages":"Pages 388-411"},"PeriodicalIF":1.1,"publicationDate":"2025-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145018511","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":"Hodge operators and groups of isometries of diagonalizable symmetric bilinear forms in characteristic two","authors":"Linus Kramer ,&nbsp;Markus J. Stroppel","doi":"10.1016/j.laa.2025.09.002","DOIUrl":"10.1016/j.laa.2025.09.002","url":null,"abstract":"<div><div>We study groups of isometries of non-alternating symmetric bilinear forms on vector spaces of characteristic two, and actions of these groups on exterior powers of the space, viewed as modules over algebras generated by Hodge operators.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"728 ","pages":"Pages 211-231"},"PeriodicalIF":1.1,"publicationDate":"2025-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145045933","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":"Integrable modules of map full Toroidal Lie algebras","authors":"Pradeep Bisht,&nbsp;Punita Batra","doi":"10.1016/j.laa.2025.08.022","DOIUrl":"10.1016/j.laa.2025.08.022","url":null,"abstract":"<div><div>In this paper, we study the irreducible objects of the category <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>f</mi><mi>i</mi><mi>n</mi></mrow></msub></math></span> of integrable representations for map full Toroidal Lie algebras with finite-dimensional weight spaces. These representations turn out to be single point evaluation modules and hence are irreducible-integrable modules for the underlying full Toroidal algebras.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"728 ","pages":"Pages 161-185"},"PeriodicalIF":1.1,"publicationDate":"2025-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145018442","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":"Norm orthogonal bases and invariants of p-adic lattices","authors":"Chi Zhang ,&nbsp;Yingpu Deng ,&nbsp;Zhaonan Wang","doi":"10.1016/j.laa.2025.09.001","DOIUrl":"10.1016/j.laa.2025.09.001","url":null,"abstract":"<div><div>In 2018, the Longest Vector Problem (LVP) and the Closest Vector Problem (CVP) in <em>p</em>-adic lattices were introduced. These problems are closely linked to the orthogonalization process. In this paper, we first prove that every <em>p</em>-adic lattice has an orthogonal basis respect to any given norm, whereas lattices in Euclidean spaces lack such bases in general. It is an improvement on Weil's result. Then, we prove that the sorted norm sequence of orthogonal basis of a <em>p</em>-adic lattice is unique and give definitions to the successive maxima and the escape distance, as the <em>p</em>-adic analogues of the successive minima and the covering radius in Euclidean lattices. Finally, we present deterministic polynomial time algorithms designed for the orthogonalization process, addressing both the LVP and the CVP with the help of an orthogonal basis of the whole vector space.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"728 ","pages":"Pages 186-210"},"PeriodicalIF":1.1,"publicationDate":"2025-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145018443","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":"Equality in some symplectic eigenvalue inequalities","authors":"Hemant K. Mishra","doi":"10.1016/j.laa.2025.08.021","DOIUrl":"10.1016/j.laa.2025.08.021","url":null,"abstract":"<div><div>In the last decade, numerous works have investigated several properties of symplectic eigenvalues. Remarkably, the results on symplectic eigenvalues have been found to be analogous to those of eigenvalues of Hermitian matrices with appropriate interpretations. In particular, symplectic analogs of famous eigenvalue inequalities are known today such as Weyl's inequalities, Lidskii's inequalities, and Schur–Horn majorization inequalities. In this paper, we provide necessary and sufficient conditions for equality in the symplectic analogs of the aforementioned inequalities.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"728 ","pages":"Pages 141-160"},"PeriodicalIF":1.1,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145010493","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":"Corrigendum to “Numerical methods for eigenvalues of singular polynomial eigenvalue problems” [Linear Algebra Appl. 719 (2025) 1–33]","authors":"Michiel E. Hochstenbach ,&nbsp;Christian Mehl ,&nbsp;Bor Plestenjak","doi":"10.1016/j.laa.2025.08.020","DOIUrl":"10.1016/j.laa.2025.08.020","url":null,"abstract":"","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"727 ","pages":"Pages 385-387"},"PeriodicalIF":1.1,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144922872","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":"Signed graphs Gσ with nullity n(Gσ)−g(Gσ)−1","authors":"Suliman Khan","doi":"10.1016/j.laa.2025.08.017","DOIUrl":"10.1016/j.laa.2025.08.017","url":null,"abstract":"<div><div>Let <span><math><msup><mrow><mi>G</mi></mrow><mrow><mi>σ</mi></mrow></msup><mo>=</mo><mo>(</mo><mi>G</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span> be a signed graph of order <span><math><mi>n</mi><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mi>σ</mi></mrow></msup><mo>)</mo></math></span>. Let denote the girth, rank, and nullity of <span><math><msup><mrow><mi>G</mi></mrow><mrow><mi>σ</mi></mrow></msup></math></span> by <span><math><mi>g</mi><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mi>σ</mi></mrow></msup><mo>)</mo></math></span>, <span><math><mi>r</mi><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mi>σ</mi></mrow></msup><mo>)</mo></math></span>, and <span><math><mi>η</mi><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mi>σ</mi></mrow></msup><mo>)</mo></math></span>, respectively. Recently, Chang and Li (2022) <span><span>[6]</span></span>, characterized connected graphs <em>G</em> with nullity <span><math><mi>n</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mi>g</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span>. In this paper, we extend the results of Chang and Li to the setting of signed graphs with some new improvements. Furthermore, we characterize signed graphs <span><math><msup><mrow><mi>G</mi></mrow><mrow><mi>σ</mi></mrow></msup></math></span> that satisfy the nullity conditions <span><math><mi>η</mi><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mi>σ</mi></mrow></msup><mo>)</mo><mo>=</mo><mi>n</mi><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mi>σ</mi></mrow></msup><mo>)</mo><mo>−</mo><mi>g</mi><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mi>σ</mi></mrow></msup><mo>)</mo></math></span> and <span><math><mi>η</mi><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mi>σ</mi></mrow></msup><mo>)</mo><mo>=</mo><mi>n</mi><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mi>σ</mi></mrow></msup><mo>)</mo><mo>−</mo><mi>g</mi><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mi>σ</mi></mrow></msup><mo>)</mo><mo>−</mo><mn>2</mn></math></span>, providing distinct characterization from those of Q. Wu et al. (2022).</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"728 ","pages":"Pages 47-62"},"PeriodicalIF":1.1,"publicationDate":"2025-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144913822","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":"Discrete quantum walks with marked vertices and their average vertex mixing matrices","authors":"Amulya Mohan,&nbsp;Hanmeng Zhan","doi":"10.1016/j.laa.2025.08.013","DOIUrl":"10.1016/j.laa.2025.08.013","url":null,"abstract":"<div><div>We study the discrete quantum walk on a regular graph <em>X</em> that assigns negative identity coins to marked vertices <em>S</em> and Grover coins to the unmarked ones. We find combinatorial bases for the eigenspaces of the transition matrix, and derive a formula for the average vertex mixing matrix <span><math><mover><mrow><mi>M</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span>.</div><div>We then find bounds for entries in <span><math><mover><mrow><mi>M</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span>, and study when these bounds are tight. In particular, the average probabilities between marked vertices are lower bounded by a matrix determined by the induced subgraph <span><math><mi>X</mi><mo>[</mo><mi>S</mi><mo>]</mo></math></span>, the vertex-deleted subgraph <span><math><mi>X</mi><mo>﹨</mo><mi>S</mi></math></span>, and the edge deleted subgraph <span><math><mi>X</mi><mo>−</mo><mi>E</mi><mo>(</mo><mi>S</mi><mo>)</mo></math></span>. We show this bound is achieved if and only if the marked vertices have walk-equitable neighborhoods in the vertex-deleted subgraph. Finally, for quantum walks attaining this bound, we determine when <span><math><mover><mrow><mi>M</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>[</mo><mi>S</mi><mo>,</mo><mi>S</mi><mo>]</mo></math></span> is symmetric, positive semidefinite or uniform.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"727 ","pages":"Pages 336-367"},"PeriodicalIF":1.1,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144904283","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 and global minimizers of the smooth stress function in Euclidean distance matrix problems","authors":"Mengmeng Song ,&nbsp;Douglas S. Gonçalves ,&nbsp;Woosuk L. Jung ,&nbsp;Carlile Lavor ,&nbsp;Antonio Mucherino ,&nbsp;Henry Wolkowicz","doi":"10.1016/j.laa.2025.08.012","DOIUrl":"10.1016/j.laa.2025.08.012","url":null,"abstract":"<div><div>We consider the nonconvex minimization problem, with quartic objective function, that arises in the exact recovery of a configuration matrix <span><math><mi>P</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>d</mi></mrow></msup></math></span> of <em>n</em> points when a Euclidean distance matrix, <strong>EDM</strong>, is given with embedding dimension <em>d</em>. It is an open question in the literature whether there are conditions such that the minimization problem admits a local nonglobal minimizer, <strong>lngm</strong>. We prove that all second-order stationary points are global minimizers whenever <span><math><mi>n</mi><mo>≤</mo><mi>d</mi><mo>+</mo><mn>1</mn></math></span>. And, for <span><math><mi>d</mi><mo>=</mo><mn>1</mn></math></span> and <span><math><mi>n</mi><mo>≥</mo><mn>7</mn><mo>&gt;</mo><mi>d</mi><mo>+</mo><mn>1</mn></math></span>, we present an example where we can analytically exhibit a local nonglobal minimizer. For more general cases, we numerically find a second-order stationary point and then prove that there indeed exists a nearby <strong>lngm</strong> for the quartic nonconvex minimization problem. Thus, we answer the previously open question about their existence in the affirmative. Our approach to finding the <strong>lngm</strong> is novel in that we first exploit the translation and rotation invariance to remove the singularities of the Hessian, and reduce the size of the problem from <em>nd</em> variables in <em>P</em> to <span><math><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo><mi>d</mi><mo>−</mo><mi>d</mi><mo>(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>2</mn></math></span> variables. This allows for stabilizing Newton's method, and for finding examples that satisfy the strict second order sufficient optimality conditions.</div><div>The motivation for being able to find global minima is to obtain <em>exact recovery</em> of the configuration matrix, even in the cases where the data is noisy and/or incomplete, without resorting to approximating solutions from convex (semidefinite programming) relaxations. In the process of our work we present new insights into when <strong>lngm</strong>s of the smooth stress function do and do not exist.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"727 ","pages":"Pages 234-267"},"PeriodicalIF":1.1,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144904382","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":"Irredundant generating sets for matrix algebras","authors":"Yonatan Blumenthal,&nbsp;Uriya A. First","doi":"10.1016/j.laa.2025.08.008","DOIUrl":"10.1016/j.laa.2025.08.008","url":null,"abstract":"<div><div>Let <em>F</em> be a field. We show that the largest irredundant generating sets for the algebra of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices over <em>F</em> have <span><math><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></math></span> elements when <span><math><mi>n</mi><mo>&gt;</mo><mn>1</mn></math></span>. (A result of Laffey states that the answer is <span><math><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></math></span> when <span><math><mi>n</mi><mo>&gt;</mo><mn>2</mn></math></span>, but its proof contains an error.) We further give a classification of the largest irredundant generating sets when <span><math><mi>n</mi><mo>∈</mo><mo>{</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>}</mo></math></span> and <em>F</em> is algebraically closed. We use this description to compute the dimension of the variety of <span><math><mo>(</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-tuples of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices which form an irredundant generating set when <span><math><mi>n</mi><mo>∈</mo><mo>{</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>}</mo></math></span>, and draw some consequences to <em>locally redundant</em> generation of Azumaya algebras. In the course of proving the classification, we also determine the largest sets <em>S</em> of subspaces of <span><math><msup><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> with the property that every <span><math><mi>V</mi><mo>∈</mo><mi>S</mi></math></span> admits a matrix stabilizing every subspace in <span><math><mi>S</mi><mo>−</mo><mo>{</mo><mi>V</mi><mo>}</mo></math></span> and not stabilizing <em>V</em>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"727 ","pages":"Pages 308-335"},"PeriodicalIF":1.1,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144904381","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}