Linear Algebra and its Applications最新文献

{"title":"Orbits under dual symplectic transvections","authors":"Jonas Sjöstrand","doi":"10.1016/j.laa.2025.02.010","DOIUrl":"10.1016/j.laa.2025.02.010","url":null,"abstract":"<div><div>Consider an arbitrary field <em>K</em> and a finite-dimensional vector space <em>X</em> over <em>K</em> equipped with a, possibly degenerate, symplectic form <em>ω</em>. Given a spanning subset <em>S</em> of <em>X</em>, for each <em>k</em> in <em>K</em> and each vector <em>s</em> in <em>S</em>, consider the symplectic transvection mapping a vector <em>x</em> to <span><math><mi>x</mi><mo>+</mo><mi>k</mi><mi>ω</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>s</mi><mo>)</mo><mi>s</mi></math></span>. The group generated by these transvections has been extensively studied, and its orbit structure is known. In this paper, we obtain corresponding results for the orbits of the dual action on <span><math><msup><mrow><mi>X</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>. As for the non-dual case, the analysis gets harder when the field contains only two elements. For that field, the dual transvection group is equivalent to a game known as the lit-only sigma game, played on a graph. Our results provide a complete solution to the reachability problem of that game, previously solved only for some special cases.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 507-530"},"PeriodicalIF":1.0,"publicationDate":"2025-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143402811","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":"Characterization of quasi-threshold graphs with two main Q-eigenvalues","authors":"Átila Jones ,&nbsp;Vilmar Trevisan ,&nbsp;Cybele T.M. Vinagre","doi":"10.1016/j.laa.2025.02.009","DOIUrl":"10.1016/j.laa.2025.02.009","url":null,"abstract":"<div><div>In this paper, we provide a structural description of certain connected cographs having <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> main signless Laplacian eigenvalues. This result allows us to characterize the cographs which are <em>quasi</em>-threshold graphs with two main <strong>Q</strong>-eigenvalues. In addition, we describe all the <em>quasi</em>-threshold graphs belonging to the subclass of generalized core-satellite graphs with <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> main <strong>Q</strong>-eigenvalues.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"711 ","pages":"Pages 68-83"},"PeriodicalIF":1.0,"publicationDate":"2025-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143427956","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 spectrum of symmetric decorated paths","authors":"Gabriel Coutinho,&nbsp;Emanuel Juliano,&nbsp;Thomás Jung Spier","doi":"10.1016/j.laa.2025.02.011","DOIUrl":"10.1016/j.laa.2025.02.011","url":null,"abstract":"<div><div>The main result of this paper states that in a rooted product of a path with rooted graphs which are disposed in a somewhat mirror-symmetric fashion, there are distinct eigenvalues supported on the end vertices of the path so that their difference is less than the square root of two in the even distance case, and less than one in the odd distance case. As a first application, we show that these end vertices cannot be involved in a quantum walk phenomenon known as perfect state transfer, significantly strengthening a recent result by two of the authors along with Godsil and van Bommel. For a second application, we show that there is no balanced integral tree of odd diameter greater than three, answering a question raised by Híc and Nedela in 1998.</div><div>Our main technique involves manipulating ratios of characteristic polynomials of graphs and subgraphs into continued fractions, and exploring in detail their analytic properties. We will also make use of a result due to Pólya and Szegö about functions that preserve the Lebesgue measure, which as far as we know is a novel application to combinatorics. In the end, we connect our machinery to a recently introduced algorithm to locate eigenvalues of trees, and with our approach we show that any graph which contains two vertices separated by a unique path that is the subdivision of a bridge with at least six inner vertices cannot be integral. As a minor corollary this implies that most trees are not integral, but we believe no one thought otherwise.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"711 ","pages":"Pages 17-39"},"PeriodicalIF":1.0,"publicationDate":"2025-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143422653","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":"Computation of an exact GCRD of several polynomial matrices: QR decomposition approach","authors":"Anjali Beniwal ,&nbsp;Tanay Saha ,&nbsp;Swanand R. Khare","doi":"10.1016/j.laa.2025.02.001","DOIUrl":"10.1016/j.laa.2025.02.001","url":null,"abstract":"<div><div>This paper addresses the problem of computing an exact Greatest Common Right Divisor (GCRD) of several univariate polynomial matrices <span><math><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>s</mi><mo>)</mo><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>(</mo><mi>s</mi><mo>)</mo></math></span>. We construct a polynomial matrix <span><math><mi>P</mi><mo>(</mo><mi>s</mi><mo>)</mo></math></span> by stacking <span><math><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>s</mi><mo>)</mo><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>(</mo><mi>s</mi><mo>)</mo></math></span>, one below the other. This results in <span><math><mi>P</mi><mo>(</mo><mi>s</mi><mo>)</mo></math></span> being wide, square or tall, each examined individually. We further prove the equivalence of rank deficiency of a particular generalized Sylvester matrix associated with <span><math><mi>P</mi><mo>(</mo><mi>s</mi><mo>)</mo></math></span> to the degree of the determinant of a GCRD of <span><math><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>s</mi><mo>)</mo><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>(</mo><mi>s</mi><mo>)</mo></math></span> when <span><math><mi>P</mi><mo>(</mo><mi>s</mi><mo>)</mo></math></span> is a tall matrix with full normal rank. This equivalence enables us to propose a method to extract a GCRD based on the ‘effectively eliminating’ <em>QR</em> (<span><math><mi>E</mi><mi>E</mi><mi>Q</mi><mi>R</mi></math></span>) decomposition of that generalized Sylvester matrix. We also propose a computationally efficient algorithm to extract the exact GCRD. To validate the theoretical findings, we provide several numerical examples.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 471-506"},"PeriodicalIF":1.0,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143402812","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":"Concrete billiard arrays of polynomial type and Leonard systems","authors":"Jimmy Vineyard","doi":"10.1016/j.laa.2025.02.006","DOIUrl":"10.1016/j.laa.2025.02.006","url":null,"abstract":"<div><div>Let <em>d</em> denote a nonnegative integer and let <span><math><mi>F</mi></math></span> denote a field. Let <em>V</em> denote a <span><math><mo>(</mo><mi>d</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-dimensional vector space over <span><math><mi>F</mi></math></span>. Given an ordering <span><math><msubsup><mrow><mo>{</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>d</mi></mrow></msubsup></math></span> of the eigenvalues of a multiplicity-free linear map <span><math><mi>A</mi><mo>:</mo><mi>V</mi><mo>→</mo><mi>V</mi></math></span>, we construct a Concrete Billiard Array <span><math><mi>L</mi></math></span> with the property that for <span><math><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>d</mi></math></span>, the <span><math><msup><mrow><mi>i</mi></mrow><mrow><mi>th</mi></mrow></msup></math></span> vector on its bottom border is in the <span><math><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>-eigenspace of <em>A</em>. The Concrete Billiard Array <span><math><mi>L</mi></math></span> is said to have polynomial type. We also show the following. Assume that there exists a Leonard system <span><math><mi>Φ</mi><mo>=</mo><mo>(</mo><mi>A</mi><mo>;</mo><msubsup><mrow><mo>{</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>d</mi></mrow></msubsup><mo>;</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>;</mo><msubsup><mrow><mo>{</mo><msubsup><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>d</mi></mrow></msubsup><mo>)</mo></math></span> where <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is the primitive idempotent of <em>A</em> corresponding to <span><math><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> for <span><math><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>d</mi></math></span>. Then, we show that after a suitable normalization, the left (resp. right) boundary of <span><math><mi>L</mi></math></span> corresponds to the Φ-split (resp. <span><math><msup><mrow><mi>Φ</mi></mrow><mrow><mo>⇓</mo></mrow></msup></math></span>-split) decomposition of <em>V</em>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 296-309"},"PeriodicalIF":1.0,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143377586","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":"A note on dual abelian varieties","authors":"Aleksandra Borówka,&nbsp;Paweł Borówka","doi":"10.1016/j.laa.2025.02.008","DOIUrl":"10.1016/j.laa.2025.02.008","url":null,"abstract":"<div><div>For any non-principal polarisation <em>D</em>, we explicitly construct <em>D</em>-polarised abelian variety <em>A</em>, such that its dual abelian variety is not (abstractly) isomorphic to <em>A</em>. For <span><math><mi>dim</mi><mo>⁡</mo><mo>(</mo><mi>A</mi><mo>)</mo><mo>&gt;</mo><mn>3</mn></math></span> the construction includes examples with submaximal Picard number equal to <span><math><msup><mrow><mo>(</mo><mi>dim</mi><mo>⁡</mo><mo>(</mo><mi>A</mi><mo>)</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>1</mn></math></span>. As a corollary, we show that a very general non-principally polarised abelian variety is not isomorphic to its dual. Moreover, we show an example of an abelian variety that is isomorphic to its dual, yet it does not admit a principal polarisation.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 458-470"},"PeriodicalIF":1.0,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143394635","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":"Piercing intersecting convex sets","authors":"Imre Bárány ,&nbsp;Travis Dillon ,&nbsp;Dömötör Pálvölgyi ,&nbsp;Dániel Varga","doi":"10.1016/j.laa.2025.02.007","DOIUrl":"10.1016/j.laa.2025.02.007","url":null,"abstract":"<div><div>Assume two finite families <span><math><mi>A</mi></math></span> and <span><math><mi>B</mi></math></span> of convex sets in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> have the property that <span><math><mi>A</mi><mo>∩</mo><mi>B</mi><mo>≠</mo><mo>∅</mo></math></span> for every <span><math><mi>A</mi><mo>∈</mo><mi>A</mi></math></span> and <span><math><mi>B</mi><mo>∈</mo><mi>B</mi></math></span>. Is there a constant <span><math><mi>γ</mi><mo>&gt;</mo><mn>0</mn></math></span> (independent of <span><math><mi>A</mi></math></span> and <span><math><mi>B</mi></math></span>) such that there is a line intersecting <span><math><mi>γ</mi><mo>|</mo><mi>A</mi><mo>|</mo></math></span> sets in <span><math><mi>A</mi></math></span> or <span><math><mi>γ</mi><mo>|</mo><mi>B</mi><mo>|</mo></math></span> sets in <span><math><mi>B</mi></math></span>? This is an intriguing Helly-type question from a paper by Martínez, Roldan and Rubin. We confirm this in the special case when all sets in <span><math><mi>A</mi></math></span> lie in parallel planes and all sets in <span><math><mi>B</mi></math></span> lie in parallel planes; in fact, one of the two families has a transversal by a single line.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 405-417"},"PeriodicalIF":1.0,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143387465","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":"Combinatorial reciprocity for Riordan arrays","authors":"Jihyeug Jang ,&nbsp;Louis W. Shapiro ,&nbsp;Minho Song","doi":"10.1016/j.laa.2025.02.002","DOIUrl":"10.1016/j.laa.2025.02.002","url":null,"abstract":"<div><div>In this paper, we develop a combinatorial reciprocity theory for Riordan arrays. Our focus is on calculating the negative territory using recurrence relations and investigating their combinatorial properties in connection with the original Riordan arrays. We take this negative part, flip it, and change signs on alternate diagonals. When this new matrix coincides with the original array, we call it a bogus-involution. These are our main focus. Noting the similarity between bogus-involutions and the previously defined pseudo-involutions, we also examine matrices that exhibit both properties. When these matrices have integer entries, as is often the case, we explore their combinatorial significance.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"721 ","pages":"Pages 419-448"},"PeriodicalIF":1.0,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144170026","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":"Subdivision and graph eigenvalues","authors":"Hitesh Kumar,&nbsp;Bojan Mohar ,&nbsp;Shivaramakrishna Pragada,&nbsp;Hanmeng Zhan","doi":"10.1016/j.laa.2025.01.044","DOIUrl":"10.1016/j.laa.2025.01.044","url":null,"abstract":"<div><div>This paper investigates the asymptotic nature of graph spectra when some edges of a graph are subdivided sufficiently many times. In the special case where all edges of a graph are subdivided, we find the exact limits of the <em>k</em>-th largest and <em>k</em>-th smallest eigenvalues for any fixed <em>k</em>. Given a graph, we show that after subdividing sufficiently many times, all but <span><math><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span> eigenvalues of the new graph will lie in the interval <span><math><mo>[</mo><mo>−</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>]</mo></math></span>. We examine the eigenvalues of the new graph outside this interval, and we prove several results that might be of independent interest.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 336-355"},"PeriodicalIF":1.0,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143378772","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":"Quantum state transfer in graphs with tails","authors":"Pierre-Antoine Bernard ,&nbsp;Christino Tamon ,&nbsp;Luc Vinet ,&nbsp;Weichen Xie","doi":"10.1016/j.laa.2025.02.005","DOIUrl":"10.1016/j.laa.2025.02.005","url":null,"abstract":"<div><div>Godsil proved that there is no quantum perfect state transfer (between vertex states) on bounded infinite graphs. We show however there exists quantum perfect state transfer in graphs with tails. The main argument used is a decoupling theorem for eventually-free Jacobi matrices (due to Golinskii). Our results rehabilitate the notion of a dark subspace which had been so far viewed in an unflattering light.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 363-384"},"PeriodicalIF":1.0,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143387464","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}