Linear Algebra and its Applications最新文献

{"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":"Matrices with exactly one real positive eigenvalue and the rest having negative or non-positive real parts","authors":"Zhibing Chen ,&nbsp;Xuerong Yong","doi":"10.1016/j.laa.2025.08.019","DOIUrl":"10.1016/j.laa.2025.08.019","url":null,"abstract":"<div><div>An <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> real or complex matrix <em>A</em> is called elliptic if it has exactly one real positive eigenvalue and all of its other eigenvalues have non-positive real parts. The real symmetric elliptic matrices were recently discussed extensively in <span><span>[4]</span></span>, <span><span>[16]</span></span>, <span><span>[19]</span></span>, <span><span>[24]</span></span> and have provided many interesting results and applications. However, when the system gets perturbed, the corresponding matrix will no longer be symmetric and such a class of matrices appears in many areas of applied mathematics and sciences. In this paper we study the general real or complex elliptic matrices. We first establish a criterion based on the Hurwitz's sequence of determinants similar to the Routh-Hurwitz's theorem on stable matrices and discuss elliptic matrices from their characteristic polynomials. We then discover that the real or complex elliptic matrices bear close relations with the <em>PN</em>-matrices and the <em>SPN</em>-matrices that appear in the trade theory of economics <span><span>[3]</span></span>, <span><span>[9]</span></span>, <span><span>[11]</span></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"728 ","pages":"Pages 121-140"},"PeriodicalIF":1.1,"publicationDate":"2025-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144989170","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":"Jordan degree type for codimension three Gorenstein algebras of small Sperner number","authors":"Nancy Abdallah ,&nbsp;Nasrin Altafi ,&nbsp;Anthony Iarrobino ,&nbsp;Joachim Yaméogo","doi":"10.1016/j.laa.2025.08.018","DOIUrl":"10.1016/j.laa.2025.08.018","url":null,"abstract":"<div><div>The Jordan type <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>ℓ</mi></mrow></msub></math></span> of a linear form <em>ℓ</em> acting on a graded Artinian algebra <em>A</em> over a field <span><math><mi>k</mi></math></span> is the partition describing the Jordan block decomposition of the multiplication map <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span>, which is nilpotent. The Jordan degree type <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>ℓ</mi></mrow></msub></math></span> is a finer invariant, describing also the initial degrees of the simple submodules of <em>A</em> in a decomposition of <em>A</em> as a direct sum of <span><math><mi>k</mi><mo>[</mo><mi>ℓ</mi><mo>]</mo></math></span>-modules. The set of Jordan types of <em>A</em> or Jordan degree types (JDT) of <em>A</em> as <em>ℓ</em> varies, is an invariant of the algebra. This invariant has been studied for codimension two graded algebras. We here extend the previous results to certain codimension three graded Artinian Gorenstein (AG) algebras - those of small Sperner number. Given a Gorenstein sequence <em>T</em> - one possible for the Hilbert function of a codimension three graded AG algebra - the irreducible variety <span><math><mrow><mi>Gor</mi></mrow><mo>(</mo><mi>T</mi><mo>)</mo></math></span> parametrizes all Gorenstein algebras of Hilbert function <em>T</em>. We here completely determine the JDT possible for all pairs <span><math><mo>(</mo><mi>A</mi><mo>,</mo><mi>ℓ</mi><mo>)</mo><mo>,</mo><mi>A</mi><mo>∈</mo><mrow><mi>Gor</mi></mrow><mo>(</mo><mi>T</mi><mo>)</mo></math></span>, for Gorenstein sequences <em>T</em> of the form <span><math><mi>T</mi><mo>=</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>,</mo><msup><mrow><mi>s</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> for Sperner number <span><math><mi>s</mi><mo>=</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn></math></span> and arbitrary multiplicity <em>k</em>. For <span><math><mi>s</mi><mo>=</mo><mn>6</mn></math></span> we delimit the prospective JDT, without verifying that each occurs.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"728 ","pages":"Pages 82-120"},"PeriodicalIF":1.1,"publicationDate":"2025-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144933354","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":"Cyclic frames in finite-dimensional Hilbert spaces","authors":"Ole Christensen ,&nbsp;Navneet Redhu ,&nbsp;Niraj K. Shukla","doi":"10.1016/j.laa.2025.08.016","DOIUrl":"10.1016/j.laa.2025.08.016","url":null,"abstract":"<div><div>Generalizing a definition by Kalra <span><span>[10]</span></span>, the purpose of this paper is to analyze cyclic frames in finite-dimensional Hilbert spaces. Cyclic frames form a subclass of the dynamical frames introduced and analyzed in detail by Aldroubi et al. in <span><span>[1]</span></span> and subsequent papers; they are particularly interesting due to their attractive properties in the context of erasure problems. By applying an alternative approach, we are able to shed new light on general dynamical frames as well as cyclic frames. In particular, we provide a characterization of dynamical frames, which in turn leads to a characterization of cyclic frames.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"728 ","pages":"Pages 63-81"},"PeriodicalIF":1.1,"publicationDate":"2025-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144926084","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 an inverse tridiagonal eigenvalue problem and its application to synchronization of network motion","authors":"Luca Dieci ,&nbsp;Cinzia Elia ,&nbsp;Alessandro Pugliese","doi":"10.1016/j.laa.2025.08.015","DOIUrl":"10.1016/j.laa.2025.08.015","url":null,"abstract":"<div><div>In this work, motivated by the study of stability of the synchronous orbit of a network with tridiagonal Laplacian matrix, we first solve an inverse eigenvalue problem which builds a tridiagonal Laplacian matrix with eigenvalues <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>0</mn><mo>&lt;</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>&lt;</mo><mo>⋯</mo><mo>&lt;</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>N</mi></mrow></msub></math></span> and null-vector <figure><img></figure>. Then, we show how this result can be used to guarantee –if possible– that a synchronous orbit of a connected tridiagonal network associated to the matrix <em>L</em> above is asymptotically stable, in the sense of having an associated negative Master Stability Function (MSF). We further show that there are limitations when we also impose symmetry for <em>L</em>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"728 ","pages":"Pages 26-46"},"PeriodicalIF":1.1,"publicationDate":"2025-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144913821","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}