Linear Algebra and its Applications最新文献

{"title":"RSMAR: An iterative method for range-symmetric linear systems","authors":"Kui Du,&nbsp;Jia-Jun Fan,&nbsp;Fang Wang","doi":"10.1016/j.laa.2025.09.024","DOIUrl":"10.1016/j.laa.2025.09.024","url":null,"abstract":"<div><div>We propose a new iterative method, named RSMAR (Range-Symmetric Minimal <strong>A</strong>-Residual), for solving range-symmetric linear systems (possibly singular). RSMAR is an extension of the MINARES method of Montoison, Orban, and Saunders [SIAM J. Matrix Anal. Appl., 46 (2025), pp. 509–529] for solving symmetric linear systems. We prove that, in exact arithmetic, RSMAR and GMRES terminate with the same (least-squares) solution when applied to range-symmetric linear systems. In cases where the reached least-squares solution is not the pseudoinverse solution, we demonstrate that a minimum-norm refinement can be used to obtain the pseudoinverse solution. We present two implementations for RSMAR. Our numerical experiments show that RSMAR outperforms GMRES on singular inconsistent range-symmetric linear systems.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"729 ","pages":"Pages 49-66"},"PeriodicalIF":1.1,"publicationDate":"2025-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145204227","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":"Projected tensor-tensor products for efficient computation of optimal multiway data representations","authors":"Katherine Keegan ,&nbsp;Elizabeth Newman","doi":"10.1016/j.laa.2025.09.018","DOIUrl":"10.1016/j.laa.2025.09.018","url":null,"abstract":"<div><div>Tensor decompositions have become essential tools for feature extraction and compression of multiway data. Recent advances in tensor operators have enabled desirable properties of standard matrix algebra to be retained for multilinear factorizations. Behind this matrix-mimetic tensor operation is an invertible matrix whose size depends quadratically on certain dimensions of the data. As a result, for large-scale multiway data, the invertible matrix can be computationally demanding to apply and invert and can lead to inefficient tensor representations in terms of construction and storage costs. In this work, we propose a new projected tensor-tensor product that relaxes the invertibility restriction to reduce computational overhead and still preserves fundamental linear algebraic properties. The transformation behind the projected product is a tall-and-skinny matrix with unitary columns, which depends only linearly on certain dimensions of the data, thereby reducing computational complexity by an order of magnitude. We provide extensive theory to prove the matrix mimeticity and the optimality of compressed representations within the projected product framework. We further prove that projected-product-based approximations outperform a comparable, non-matrix-mimetic tensor factorization. We support the theoretical findings and demonstrate the practical benefits of projected products through numerical experiments on video, hyperspectral imaging, synthetic, and dynamical systems data. All code for this paper is available at <span><span>https://github.com/elizabethnewman/projected-products.git</span><svg><path></path></svg></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"729 ","pages":"Pages 100-147"},"PeriodicalIF":1.1,"publicationDate":"2025-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145204473","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":"Error analysis of randomized symplectic model order reduction for Hamiltonian systems","authors":"Robin Herkert ,&nbsp;Patrick Buchfink ,&nbsp;Bernard Haasdonk ,&nbsp;Johannes Rettberg ,&nbsp;Jörg Fehr","doi":"10.1016/j.laa.2025.09.022","DOIUrl":"10.1016/j.laa.2025.09.022","url":null,"abstract":"<div><div>Solving high-dimensional dynamical systems in multi-query or real-time applications requires efficient surrogate modelling techniques, as e.g., achieved via model order reduction (MOR). If these systems are Hamiltonian systems their physical structure should be preserved during the reduction, which can be ensured by applying symplectic basis generation techniques such as the complex SVD (cSVD). Recently, randomized symplectic methods such as the randomized complex singular value decomposition (rcSVD) have been developed for a more efficient computation of symplectic bases that preserve the Hamiltonian structure during MOR. In the current paper, we present two error bounds for the rcSVD basis depending on the choice of hyperparameters. We provide numerical experiments that demonstrate the efficiency of randomized symplectic basis generation and compare the bounds numerically.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"729 ","pages":"Pages 67-99"},"PeriodicalIF":1.1,"publicationDate":"2025-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145204228","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 spectral reconstruction problem revisited","authors":"Weifang Lv ,&nbsp;Wei Wang ,&nbsp;Wei Wang ,&nbsp;Hao Zhang","doi":"10.1016/j.laa.2025.09.020","DOIUrl":"10.1016/j.laa.2025.09.020","url":null,"abstract":"<div><div>This paper focuses on the following spectral reconstruction problem: Can a graph be uniquely determined (up to isomorphism) by the collection of its spectrum and the spectra of its vertex-deleted graphs? We say two graphs are <em>hyper-cospectral</em> if they share identical spectrum and identical spectra for their vertex-deleted subgraphs. A graph <em>G</em> is <em>spectrally reconstructible</em> (SRC for short) if any graph <em>H</em> that is hyper-cospectral with <em>G</em> is also isomorphic to <em>G</em>. Tutte <span><span>[13]</span></span> showed that graphs with irreducible characteristic polynomials are SRC. We aim to extend this result to a larger family of graphs known as controllable graphs. Since not all controllable graphs are SRC, we address the problem: Which controllable graphs are SRC? We provide a proof for a family of controllable bipartite graphs whose characteristic polynomials have exactly two irreducible factors. Moreover, we demonstrate that for a non-controllable graph <em>G</em> with a characteristic polynomial having exactly two irreducible factors, if a hyper-cospectral graph exists, then their complements are also hyper-cospectral. In addition, we also present an algorithm for determining the spectral reconstructibility of graphs whose characteristic polynomials split into exactly two irreducible factors, as well as for finding their hyper-cospectral mates when they exist. Some examples are also provided to illustrate our results.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"729 ","pages":"Pages 1-23"},"PeriodicalIF":1.1,"publicationDate":"2025-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145204225","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":"Maximum spread of graphs without paths of specified length","authors":"Wenyan Wang,&nbsp;Yi Wang","doi":"10.1016/j.laa.2025.09.019","DOIUrl":"10.1016/j.laa.2025.09.019","url":null,"abstract":"<div><div>The spread of a graph is defined as the difference between the largest and smallest eigenvalues of its adjacency matrix. In this paper, we investigate extremal problems for the spread of graphs that forbid paths of a specified length. For <span><math><mi>k</mi><mo>≥</mo><mn>1</mn></math></span> and <em>n</em> sufficiently large, we show that the <em>n</em>-vertex <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>2</mn></mrow></msub></math></span>-free graph achieving maximum spread is the join of a <em>k</em>-vertex clique and an independent set of <span><math><mi>n</mi><mo>−</mo><mi>k</mi></math></span> vertices. We also show that the extremal graph for <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>3</mn></mrow></msub></math></span>-free graphs:<ul><li><span>•</span><span><div>For <span><math><mi>k</mi><mo>∈</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>}</mo></math></span>, the spread is maximized by the join of a <em>k</em>-vertex clique with the disjoint union of an edge and an independent set of <span><math><mi>n</mi><mo>−</mo><mi>k</mi><mo>−</mo><mn>2</mn></math></span> vertices.</div></span></li><li><span>•</span><span><div>For <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span>, the spread is maximized by the join of a <em>k</em>-vertex clique with an independent set with <span><math><mi>n</mi><mo>−</mo><mi>k</mi></math></span> vertices.</div></span></li></ul></div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"729 ","pages":"Pages 24-48"},"PeriodicalIF":1.1,"publicationDate":"2025-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145204226","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":"Bilinear maps having Jordan product property","authors":"Jorge J. Garcés ,&nbsp;Mykola Khrypchenko","doi":"10.1016/j.laa.2025.09.013","DOIUrl":"10.1016/j.laa.2025.09.013","url":null,"abstract":"<div><div>We study symmetric continuous bilinear maps <em>V</em> on a C^⁎ -algebra <em>A</em> that have the Jordan product property at a fixed element <span><math><mi>z</mi><mo>∈</mo><mi>A</mi></math></span>. We show that, whenever <em>A</em> is a finite direct sum of infinite simple von Neumann algebras, such a map <em>V</em> has the square-zero property. Then, it is proved that <span><math><mi>V</mi><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo><mo>=</mo><mi>T</mi><mo>(</mo><mi>a</mi><mo>∘</mo><mi>b</mi><mo>)</mo></math></span> for some bounded linear map <em>T</em> on <em>A</em>. As a consequence, Jordan homomorphisms and derivations at <span><math><mi>z</mi><mo>∈</mo><mi>A</mi></math></span> are characterized.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"728 ","pages":"Pages 435-448"},"PeriodicalIF":1.1,"publicationDate":"2025-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145155058","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 complete version of Kursov's theorem for matrices over division rings","authors":"Tran Nam Son","doi":"10.1016/j.laa.2025.09.015","DOIUrl":"10.1016/j.laa.2025.09.015","url":null,"abstract":"<div><div>Let <em>D</em> be a division ring with center <em>F</em>, and let <span><math><mi>n</mi><mo>&gt;</mo><mn>1</mn></math></span> be an integer. A known result due to Kursov asserts that if <em>D</em> is finite-dimensional over <em>F</em>, then the commutator width of the general linear group <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>D</mi><mo>)</mo></math></span> is at most one greater than that of <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>D</mi><mo>)</mo></math></span>. In the absence of the finite-dimensionality assumption, recent research has made significant progress, though the developments typically cease once <em>F</em> is infinite or <em>D</em> is algebraic over <em>F</em>. The purpose of this paper is to show that these restrictions are, in fact, unnecessary.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"728 ","pages":"Pages 376-382"},"PeriodicalIF":1.1,"publicationDate":"2025-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145118082","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 factorability of infinite graphs","authors":"Babak Miraftab ,&nbsp;Heydar Radjavi ,&nbsp;Sho Suda","doi":"10.1016/j.laa.2025.09.011","DOIUrl":"10.1016/j.laa.2025.09.011","url":null,"abstract":"<div><div>A graph <em>G</em> is said to be <em>factorized</em> into graphs <em>H</em> and <em>K</em> via a matrix product if there exist adjacency matrices <em>A</em>, <em>B</em>, and <em>C</em> for <em>G</em>, <em>H</em>, and <em>K</em>, respectively, such that <span><math><mi>A</mi><mo>=</mo><mi>B</mi><mi>C</mi></math></span>. Recently, Maghsoudi et al. proved that the complete graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> admits a factorization if and only if <span><math><mi>n</mi><mo>=</mo><mn>4</mn><mi>k</mi><mo>+</mo><mn>1</mn></math></span>. In this note, we show that, in contrast to the finite case, the (countably) infinite complete graph admits a factorization via a matrix product. In addition, they showed that the complete bipartite graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span> has a factorization if and only if both <em>m</em> and <em>n</em> are even. We extend this result to the infinite complete bipartite graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span>, namely: the infinite complete bipartite graph admits a factorization if and only if the size of the finite part (if it exists) is even. Finally, we show that the <em>n</em>-dimensional grid admits a factorization for all <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"728 ","pages":"Pages 409-418"},"PeriodicalIF":1.1,"publicationDate":"2025-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145118084","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 socle of a class of Steinberg algebras","authors":"Lisa Orloff Clark ,&nbsp;Cristóbal Gil Canto ,&nbsp;Dolores Martín Barquero ,&nbsp;Cándido Martín González ,&nbsp;Iván Ruiz Campos","doi":"10.1016/j.laa.2025.09.016","DOIUrl":"10.1016/j.laa.2025.09.016","url":null,"abstract":"<div><div>We study minimal left ideals in Steinberg algebras of Hausdorff groupoids. We establish a relationship between minimal left ideals in the algebra and open singletons in the unit space of the groupoid. We apply this to obtain results about the socle of Steinberg algebras under certain hypotheses. This encompasses known results about Leavitt path algebras and improves on Kumjian-Pask algebra results to include higher-rank graphs that are not row-finite.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"728 ","pages":"Pages 449-464"},"PeriodicalIF":1.1,"publicationDate":"2025-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145154438","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 upper bound of multiplicity of (signless) Laplacian eigenvalue two","authors":"Xueying Li ,&nbsp;Ji-Ming Guo ,&nbsp;Fenglei Tian ,&nbsp;Zhiwen Wang","doi":"10.1016/j.laa.2025.09.012","DOIUrl":"10.1016/j.laa.2025.09.012","url":null,"abstract":"<div><div>For a graph <em>G</em>, let <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>L</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>,</mo><mn>2</mn><mo>)</mo></math></span> (resp., <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>Q</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>,</mo><mn>2</mn><mo>)</mo></math></span>) denote the multiplicity of Laplacian (resp., signless Laplacian) eigenvalue 2 of <em>G</em>. Wang et al. (2021) <span><span>[18]</span></span> proved that <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>L</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>,</mo><mn>2</mn><mo>)</mo><mo>≤</mo><mi>c</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mn>1</mn></math></span> for a connected graph <em>G</em>, where <span><math><mi>c</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is the cyclomatic number of <em>G</em>. Very recently, Zhao and Yu (2025) <span><span>[19]</span></span> proved that <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>Q</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>,</mo><mn>2</mn><mo>)</mo><mo>≤</mo><mi>c</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mn>1</mn></math></span> for a connected graph with a perfect matching. Let <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the even cyclomatic number of <em>G</em>, defined as the minimum number of edges whose deletion eliminates all even cycles in <em>G</em>. In this paper, for a connected graph <em>G</em>, we prove that<span><span><span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>L</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>,</mo><mn>2</mn><mo>)</mo><mo>≤</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mn>1</mn><mspace></mspace><mtext>and</mtext><mspace></mspace><msub><mrow><mi>m</mi></mrow><mrow><mi>Q</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>,</mo><mn>2</mn><mo>)</mo><mo>≤</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mn>1</mn><mo>,</mo></math></span></span></span> improving the two aforementioned results since <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>c</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"728 ","pages":"Pages 419-434"},"PeriodicalIF":1.1,"publicationDate":"2025-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145155057","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}