{"title":"Rosenbrock's theorem on system matrices over elementary divisor domains","authors":"Froilán M. Dopico , Vanni Noferini , Ion Zaballa","doi":"10.1016/j.laa.2025.01.028","DOIUrl":"10.1016/j.laa.2025.01.028","url":null,"abstract":"<div><div>Rosenbrock's theorem on polynomial system matrices is a classical result in linear systems theory that relates the Smith-McMillan form of a rational matrix <em>G</em> with the Smith form of an irreducible polynomial system matrix <em>P</em> giving rise to <em>G</em> and the Smith form of a submatrix of <em>P</em>. This theorem has been essential in the development of algorithms for computing the poles and zeros of a rational matrix via linearizations and generalized eigenvalue algorithms. In this paper, we extend Rosenbrock's theorem to system matrices <em>P</em> with entries in an arbitrary elementary divisor domain <span><math><mi>R</mi></math></span> and matrices <em>G</em> with entries in the field of fractions of <span><math><mi>R</mi></math></span>. These are the most general rings where the involved Smith-McMillan and Smith forms both exist and, so, where the problem makes sense. Moreover, we analyze in detail what happens when the system matrix is not irreducible. Finally, we explore how Rosenbrock's theorem can be extended when the system matrix <em>P</em> itself has entries in the field of fractions of the elementary divisor domain.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 10-49"},"PeriodicalIF":1.0,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143138080","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":"When does subtracting a rank-one approximation decrease tensor rank?","authors":"Emil Horobeţ , Ettore Teixeira Turatti","doi":"10.1016/j.laa.2025.01.025","DOIUrl":"10.1016/j.laa.2025.01.025","url":null,"abstract":"<div><div>Subtracting a critical rank-one approximation from a matrix always results in a matrix with a lower rank. This is not true for tensors in general. Motivated by this, we ask the question: what is the closure of the set of those tensors for which subtracting some of its critical rank-one approximation from it and repeating the process we will eventually get to zero? In this article, we show how to construct this variety of tensors and we show how this is connected to the bottleneck points of the variety of rank-one tensors (and in general to the singular locus of the hyperdeterminant), and how this variety can be equal to and in some cases be more than (weakly) orthogonally decomposable tensors.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"709 ","pages":"Pages 397-415"},"PeriodicalIF":1.0,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143129888","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":"On the nullity of middle graphs","authors":"Xinmei Yuan , Danyi Li , Weigen Yan","doi":"10.1016/j.laa.2025.01.030","DOIUrl":"10.1016/j.laa.2025.01.030","url":null,"abstract":"<div><div>Let <em>G</em> be a connected graph, and let <span><math><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and <span><math><mi>M</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the line graph and middle graph of <em>G</em>. Gutman and Sciriha (On the nullity of line graphs of trees, Discrete Mathematics, 232 (2001), 35-45) proved that the nullity <span><math><mi>η</mi><mo>(</mo><mi>L</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>)</mo></math></span> of <span><math><mi>L</mi><mo>(</mo><mi>T</mi><mo>)</mo></math></span> of a tree <em>T</em> satisfies <span><math><mi>η</mi><mo>(</mo><mi>L</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>)</mo><mo>=</mo><mn>0</mn></math></span> or <span><math><mi>η</mi><mo>(</mo><mi>L</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>)</mo><mo>=</mo><mn>1</mn></math></span>. But the problem to determine which trees <em>T</em> satisfy <span><math><mi>η</mi><mo>(</mo><mi>L</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>)</mo><mo>=</mo><mn>0</mn></math></span> or <span><math><mi>η</mi><mo>(</mo><mi>L</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>)</mo><mo>=</mo><mn>1</mn></math></span> is still open. In this paper, we prove that <span><math><mi>η</mi><mo>(</mo><mi>M</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>=</mo><mn>1</mn></math></span> if <em>G</em> is a bipartite graph, and <span><math><mi>η</mi><mo>(</mo><mi>M</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>=</mo><mn>0</mn></math></span> otherwise. As an application, we show that <span><math><mi>η</mi><mo>(</mo><mi>G</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo>)</mo><mo>)</mo><mo>=</mo><mn>1</mn></math></span> for the so-called silicate network <span><math><mi>G</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo>)</mo></math></span> obtained from the hexagonal lattice in the context of statistical physics.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 1-9"},"PeriodicalIF":1.0,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143138520","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 dimension formula for certain twisted Jacquet modules of a cuspidal representation of GL(n,Fq)","authors":"Kumar Balasubramanian , Himanshi Khurana","doi":"10.1016/j.laa.2025.01.027","DOIUrl":"10.1016/j.laa.2025.01.027","url":null,"abstract":"<div><div>Let <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span> be a positive integer. Let <em>F</em> be the finite field of order <em>q</em> and <span><math><mi>G</mi><mo>=</mo><mi>GL</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span>. Let <span><math><mi>P</mi><mo>=</mo><mi>M</mi><mi>N</mi></math></span> be the standard parabolic subgroup of <em>G</em> corresponding to the partition <span><math><mo>(</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo>)</mo></math></span>. Let <span><math><mi>A</mi><mo>∈</mo><mi>M</mi><mo>(</mo><mo>(</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo>)</mo><mo>×</mo><mi>k</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> be a rank <em>t</em> matrix. In this paper, we compute the dimension formula for the twisted Jacquet module <span><math><msub><mrow><mi>π</mi></mrow><mrow><mi>N</mi><mo>,</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mi>A</mi></mrow></msub></mrow></msub></math></span> that depends on <span><math><mi>n</mi><mo>,</mo><mi>k</mi></math></span> and <em>t</em>, when <em>π</em> is an irreducible cuspidal representation of <em>G</em> and <span><math><msub><mrow><mi>ψ</mi></mrow><mrow><mi>A</mi></mrow></msub></math></span> is a character of <em>N</em> associated with <em>A</em>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 151-164"},"PeriodicalIF":1.0,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143138084","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":"Multiple typical ranks in matrix completion","authors":"Mareike Dressler , Robert Krone","doi":"10.1016/j.laa.2025.01.026","DOIUrl":"10.1016/j.laa.2025.01.026","url":null,"abstract":"<div><div>Low-rank matrix completion addresses the problem of completing a matrix from a certain set of generic specified entries. Over the complex numbers a matrix with a given entry pattern can be uniquely completed to a specific rank, called the generic completion rank. Completions over the reals may generically have multiple completion ranks, called typical ranks. We demonstrate techniques for proving that many sets of specified entries have only one typical rank, and show other families with two typical ranks, specifically focusing on entry sets represented by circulant graphs. This generalizes the results of Bernstein, Blekherman, and Sinn. In particular, we provide a complete characterization of the set of unspecified entries of an <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrix such that <span><math><mi>n</mi><mo>−</mo><mn>1</mn></math></span> is a typical rank and fully determine the typical ranks of an <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrix with unspecified diagonal for <span><math><mi>n</mi><mo><</mo><mn>9</mn></math></span>. Moreover, we study the asymptotic behavior of typical ranks and present results regarding unique matrix completions.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 165-182"},"PeriodicalIF":1.0,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143277786","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}
Bojan Kuzma , Chi-Kwong Li , Edward Poon , Sushil Singla
{"title":"Linear preservers of parallel matrix pairs with respect to the k-numerical radius","authors":"Bojan Kuzma , Chi-Kwong Li , Edward Poon , Sushil Singla","doi":"10.1016/j.laa.2025.01.019","DOIUrl":"10.1016/j.laa.2025.01.019","url":null,"abstract":"<div><div>Let <span><math><mn>1</mn><mo>≤</mo><mi>k</mi><mo><</mo><mi>n</mi></math></span> be integers. Two <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices <em>A</em> and <em>B</em> form a parallel pair with respect to the <em>k</em>-numerical radius <span><math><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> if <span><math><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>+</mo><mi>μ</mi><mi>B</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>B</mi><mo>)</mo></math></span> for some scalar <em>μ</em> with <span><math><mo>|</mo><mi>μ</mi><mo>|</mo><mo>=</mo><mn>1</mn></math></span>; they form a TEA (triangle equality attaining) pair if the preceding equation holds for <span><math><mi>μ</mi><mo>=</mo><mn>1</mn></math></span>. We classify linear bijections on <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and on <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> which preserve parallel pairs or TEA pairs. Such preservers are scalar multiples of <span><math><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>-isometries, except for some exceptional maps on <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> when <span><math><mi>n</mi><mo>=</mo><mn>2</mn><mi>k</mi></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"709 ","pages":"Pages 342-363"},"PeriodicalIF":1.0,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143129883","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":"Minimal error momentum Bregman-Kaczmarz","authors":"Dirk A. Lorenz, Maximilian Winkler","doi":"10.1016/j.laa.2025.01.024","DOIUrl":"10.1016/j.laa.2025.01.024","url":null,"abstract":"<div><div>The Bregman-Kaczmarz method is an iterative method which can solve strongly convex problems with linear constraints and uses only one or a selected number of rows of the system matrix in each iteration, thereby making it amenable for large-scale systems. To speed up convergence, we investigate acceleration by heavy ball momentum in the so-called dual update. Heavy ball acceleration of the Kaczmarz method with constant parameters has turned out to be difficult to analyze, in particular no accelerated convergence for the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-error of the iterates has been proven to the best of our knowledge. Here we propose a way to adaptively choose the momentum parameter by a minimal-error principle similar to a recently proposed method for the standard randomized Kaczmarz method. The momentum parameter can be chosen to exactly minimize the error in the next iterate or to minimize a relaxed version of the minimal error principle. The former choice leads to a theoretically optimal step while the latter is cheaper to compute. We prove improved convergence results compared to the non-accelerated method. Numerical experiments show that the proposed methods can accelerate convergence in practice, also for matrices which arise from applications such as computational tomography.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"709 ","pages":"Pages 416-448"},"PeriodicalIF":1.0,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143129886","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 Aα-spectral radius of {C(3,3),C(4,3)}-free graphs","authors":"S. Pirzada, Amir Rehman","doi":"10.1016/j.laa.2025.01.023","DOIUrl":"10.1016/j.laa.2025.01.023","url":null,"abstract":"<div><div>For a simple graph <em>G</em> and for any <span><math><mi>α</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span>, Nikiforov defined the generalized adjacency matrix as <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>α</mi><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mo>(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo>)</mo><mi>A</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, where <span><math><mi>A</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and <span><math><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> are the adjacency and degree diagonal matrices of <em>G</em>, respectively. The largest eigenvalue of <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is called the generalized adjacency spectral radius (or <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>-spectral radius) of <em>G</em>. Let <span><math><mi>C</mi><mo>(</mo><mi>l</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> denote the graph obtained from <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>l</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> by superimposing an edge of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>l</mi></mrow></msub></math></span> with an edge of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span>. If a graph is free of both <span><math><mi>C</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>3</mn><mo>)</mo></math></span> and <span><math><mi>C</mi><mo>(</mo><mn>4</mn><mo>,</mo><mn>3</mn><mo>)</mo></math></span>, we call it a <span><math><mo>{</mo><mi>C</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>,</mo><mi>C</mi><mo>(</mo><mn>4</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>}</mo></math></span>-free graph. In this paper, we give a sharp upper bound on the <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>-spectral radius of <span><math><mo>{</mo><mi>C</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>,</mo><mi>C</mi><mo>(</mo><mn>4</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>}</mo></math></span>-free graphs for <span><math><mi>α</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></math></span>. We show that the extremal graph attaining the bound is the 2-partite Turán graph.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"709 ","pages":"Pages 385-396"},"PeriodicalIF":1.0,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143129893","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":"Some families of digraphs determined by the complementarity spectrum","authors":"Diego Bravo , Florencia Cubría , Marcelo Fiori , Gustavo Rama","doi":"10.1016/j.laa.2025.01.022","DOIUrl":"10.1016/j.laa.2025.01.022","url":null,"abstract":"<div><div>In this paper we study seven families of digraphs, and we determine whether digraphs in these families can be determined by their spectral radius. These seven families have been characterized as the only families of digraphs with exactly three complementarity eigenvalues <span><span>[1]</span></span>, and therefore our results have consequences in this context, showing which families can be determined by the complementarity spectrum. As a particular case, we prove that the <em>θ</em>-digraphs can be characterized by the spectral radius, extending some recent results on this family <span><span>[2]</span></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"709 ","pages":"Pages 364-384"},"PeriodicalIF":1.0,"publicationDate":"2025-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143129895","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":"Additive maps preserving rank-bounded sets of matrices","authors":"E. Akhmedova , A. Guterman , I. Spiridonov","doi":"10.1016/j.laa.2025.01.018","DOIUrl":"10.1016/j.laa.2025.01.018","url":null,"abstract":"<div><div>Let <span><math><mn>2</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></math></span> be integers and <span><math><msub><mrow><mi>Mat</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math></span> be the linear space of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices over a field <span><math><mi>F</mi></math></span> of characteristic different from 2. Denote by <span><math><msup><mrow><mi>Γ</mi></mrow><mrow><mo>≥</mo><mi>k</mi></mrow></msup></math></span> the set of matrices in <span><math><msub><mrow><mi>Mat</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math></span> of rank greater than or equal to <em>k</em>. The main goal of the present paper is to obtain a characterization of additive maps <span><math><mi>f</mi><mo>:</mo><msub><mrow><mi>Mat</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo><mo>→</mo><msub><mrow><mi>Mat</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math></span> satisfying <span><math><mi>f</mi><mo>(</mo><msup><mrow><mi>Γ</mi></mrow><mrow><mo>≥</mo><mi>k</mi></mrow></msup><mo>)</mo><mo>=</mo><msup><mrow><mi>Γ</mi></mrow><mrow><mo>≥</mo><mi>k</mi></mrow></msup></math></span> with either <span><math><mi>n</mi><mo><</mo><mn>2</mn><mi>k</mi><mo>−</mo><mn>2</mn></math></span> or <span><math><mi>F</mi></math></span> has characteristic <span><math><mrow><mi>char</mi><mspace></mspace></mrow><mo>(</mo><mi>F</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span> or <span><math><mrow><mi>char</mi><mspace></mspace></mrow><mo>(</mo><mi>F</mi><mo>)</mo><mo>≥</mo><mi>k</mi></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"709 ","pages":"Pages 331-341"},"PeriodicalIF":1.0,"publicationDate":"2025-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143129885","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}
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