{"title":"On the largest singular vector of the Redheffer matrix","authors":"François Clément, Stefan Steinerberger","doi":"10.1016/j.laa.2025.07.003","DOIUrl":"10.1016/j.laa.2025.07.003","url":null,"abstract":"<div><div>The Redheffer matrix <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msup></math></span> is defined by setting <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mn>1</mn></math></span> if <span><math><mi>j</mi><mo>=</mo><mn>1</mn></math></span> or <em>i</em> divides <em>j</em> and 0 otherwise. One of its many interesting properties is that <span><math><mi>det</mi><mo></mo><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>=</mo><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn><mo>+</mo><mi>ε</mi></mrow></msup><mo>)</mo></math></span> is equivalent to the Riemann hypothesis. The singular vector <span><math><mi>v</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> corresponding to the largest singular value carries a lot of information about the prime factorization of the integers: <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> is small if <em>k</em> is prime and large if <em>k</em> has many divisors. We prove that the vector <em>w</em> whose <em>k</em>-th entry is the sum of the inverse divisors of <em>k</em>, <span><math><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>d</mi><mo>|</mo><mi>k</mi></mrow></msub><mn>1</mn><mo>/</mo><mi>d</mi></math></span>, is close to a singular vector in a precise quantitative sense.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"725 ","pages":"Pages 96-114"},"PeriodicalIF":1.0,"publicationDate":"2025-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144597296","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":"Frobenius extensions about centralizer matrix algebras","authors":"Qikai Wang, Haiyan Zhu","doi":"10.1016/j.laa.2025.07.002","DOIUrl":"10.1016/j.laa.2025.07.002","url":null,"abstract":"<div><div>This paper investigates the conditions under which the centralizer algebra <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>c</mi><mo>,</mo><mi>R</mi><mo>)</mo></math></span> of a matrix <span><math><mi>c</mi><mo>∈</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span> is a (separable) Frobenius extension of the algebra <em>R</em>. For an algebra <em>R</em> over an integral domain <span><math><mi>k</mi></math></span>, we provide necessary and sufficient conditions for <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>c</mi><mo>,</mo><mi>R</mi><mo>)</mo><mo>/</mo><mi>R</mi></math></span> to be a (separable) Frobenius extension when <em>c</em> is in Jordan canonical form with eigenvalues in <span><math><mi>k</mi></math></span>. We extend this analysis to arbitrary matrices over a field and derive conditions for matrix diagonalizability through Frobenius extensions.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"725 ","pages":"Pages 18-37"},"PeriodicalIF":1.0,"publicationDate":"2025-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144580460","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":"Sharp upper bounds on the second largest signless Laplacian eigenvalues of connected graphs","authors":"Shu-Guang Guo, Rong Zhang","doi":"10.1016/j.laa.2025.07.004","DOIUrl":"10.1016/j.laa.2025.07.004","url":null,"abstract":"<div><div>Let <em>G</em> be a connected graph with <em>n</em> vertices and <em>m</em> edges, and <span><math><msub><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> denote the second largest signless Laplacian eigenvalue of <em>G</em>. A conjecture, due to Cvetković et al. (2007), asserts that <span><math><msub><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>n</mi><mo>−</mo><mn>6</mn><mo>+</mo><mo>(</mo><mn>2</mn><mi>m</mi><mo>+</mo><mn>8</mn><mo>)</mo><mo>/</mo><mi>n</mi></math></span> with equality if and only if <em>G</em> is the complete bipartite graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn><mo>,</mo><mspace></mspace><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub></math></span>. In this paper, we give sharp upper bounds on <span><math><msub><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> for a connected (minimally 2-connected) graph with given size. Employing the upper bounds, we prove that the conjecture holds for a connected bipartite graph, for a minimally 2-connected graph and for a connected graph with <span><math><mi>m</mi><mo>≠</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>5</mn></math></span> and <span><math><mn>2</mn><mi>n</mi><mo>−</mo><mn>6</mn></math></span>, respectively.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"725 ","pages":"Pages 70-95"},"PeriodicalIF":1.0,"publicationDate":"2025-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144597295","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":"Universal winners in trees","authors":"Sirshendu Pan , Steve Kirkland , Sukanta Pati","doi":"10.1016/j.laa.2025.06.023","DOIUrl":"10.1016/j.laa.2025.06.023","url":null,"abstract":"<div><div>Let <span><math><mi>A</mi><mo>∈</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> be nonnegative, irreducible and <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi></mrow></msub><msubsup><mrow><mi>e</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi></mrow></msubsup></math></span>, where <span><math><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is the <em>i</em>-th standard basis vector. For a fixed <span><math><msub><mrow><mi>t</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>></mo><mn>0</mn></math></span>, an index <span><math><mi>p</mi><mo>∈</mo><mo>[</mo><mi>n</mi><mo>]</mo><mo>:</mo><mo>=</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span> is called a winner for the value <span><math><msub><mrow><mi>t</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> if the spectral radius <span><math><mi>ρ</mi><mo>(</mo><mi>A</mi><mo>+</mo><msub><mrow><mi>t</mi></mrow><mrow><mn>0</mn></mrow></msub><msub><mrow><mi>E</mi></mrow><mrow><mi>p</mi><mi>p</mi></mrow></msub><mo>)</mo><mo>=</mo><munder><mi>max</mi><mrow><mi>i</mi><mo>∈</mo><mo>[</mo><mi>n</mi><mo>]</mo></mrow></munder><mo></mo><mi>ρ</mi><mo>(</mo><mi>A</mi><mo>+</mo><msub><mrow><mi>t</mi></mrow><mrow><mn>0</mn></mrow></msub><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi><mi>i</mi></mrow></msub><mo>)</mo></math></span>. If <em>p</em> remains a winner for each <span><math><mi>t</mi><mo>></mo><mn>0</mn></math></span>, then it is called a universal winner. The concepts have been introduced in 1996 and studied in only a few articles till now. When <em>G</em> is a simple connected graph (or a strongly connected digraph), the nonnegative weighted adjacency matrix <span><math><mi>A</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> being irreducible, one can talk of a universal winner vertex with respect to <span><math><mi>A</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. The universal winners seem to capture the graph structures well. It is known that the only connected digraph <em>G</em> in which all vertices are universal winners with respect to all nonnegative weighted <span><math><mi>A</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, is the directed cycle, thereby characterizing it. Let <span><math><mi>U</mi><mo>⊂</mo><mo>[</mo><mi>n</mi><mo>]</mo></math></span> be nonempty. In a recent article, the class of directed connected graphs with vertex set <span><math><mo>[</mo><mi>n</mi><mo>]</mo></math></span>, for which only the vertices in <em>U</em> are the universal winners with respect to all nonnegative weighted <span><math><mi>A</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> was characterized, generalizing the earlier result. Many other combinatorial results exploiting the graph structure were proved establishing the importance of the study of universal winner vertices. In this article, ","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"725 ","pages":"Pages 38-69"},"PeriodicalIF":1.0,"publicationDate":"2025-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144588361","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":"Mutually orthogonal unitary and orthogonal matrices","authors":"Zhiwei Song, Lin Chen, Saiqi Liu","doi":"10.1016/j.laa.2025.06.025","DOIUrl":"10.1016/j.laa.2025.06.025","url":null,"abstract":"<div><div>We introduce the concept of order-<em>d n</em>-OU and <em>n</em>-OO sets, which consist of <em>n</em> mutually orthogonal order-<em>d</em> unitary and real orthogonal matrices under Hilbert-Schmidt inner product. We show that for arbitrary <em>d</em>, there exists order-<em>d</em> <span><math><msup><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-OU set. However, real orthogonal matrices show strict limits, as we prove that an order-three <em>n</em>-OO set exists only if <span><math><mi>n</mi><mo>≤</mo><mn>4</mn></math></span>. As an application in quantum information theory, we establish that the maximum number of unextendible maximally entangled bases within a real two-qutrit system is four. Further, we propose a new matrix decomposition approach, defining an <em>n</em>-OU (resp. <em>n</em>-OO) decomposition for a matrix as a linear combination of <em>n</em> matrices from an <em>n</em>-OU (resp. <em>n</em>-OO) set. We show that any order-<em>d</em> matrix has a <em>d</em>-OU decomposition. As contrast, we prove the existence of real matrices that do not possess any <em>n</em>-OO decomposition by providing explicit criteria for an order-three real matrix to have an <em>n</em>-OO decomposition.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"725 ","pages":"Pages 1-17"},"PeriodicalIF":1.0,"publicationDate":"2025-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144570121","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":"Simultaneous direct sum decompositions of several multivariate polynomials","authors":"Lishan Fang, Hua-Lin Huang, Lili Liao","doi":"10.1016/j.laa.2025.06.022","DOIUrl":"10.1016/j.laa.2025.06.022","url":null,"abstract":"<div><div>We consider the problem of simultaneous direct sum decomposition of a set of multivariate polynomials. To this end, we extend Harrison's center theory for a single homogeneous polynomial to this broader setting. It is shown that the center of a set of polynomials is a special Jordan algebra, and simultaneous direct sum decompositions of the given polynomials are in bijection with complete sets of orthogonal idempotents of their center algebra. Several examples are provided to illustrate the performance of this method.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"724 ","pages":"Pages 320-335"},"PeriodicalIF":1.0,"publicationDate":"2025-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144563447","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}
Alexandru Chirvasitu , Ilja Gogić , Mateo Tomašević
{"title":"A variant of Šemrl's preserver theorem for singular matrices","authors":"Alexandru Chirvasitu , Ilja Gogić , Mateo Tomašević","doi":"10.1016/j.laa.2025.06.021","DOIUrl":"10.1016/j.laa.2025.06.021","url":null,"abstract":"<div><div>For positive integers <span><math><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></math></span> let <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> be the algebra of all <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> complex matrices and <span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>≤</mo><mi>k</mi></mrow></msubsup></math></span> its subset consisting of all matrices of rank at most <em>k</em>. We first show that whenever <span><math><mi>k</mi><mo>></mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>, any continuous spectrum-shrinking map <span><math><mi>ϕ</mi><mo>:</mo><msubsup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>≤</mo><mi>k</mi></mrow></msubsup><mo>→</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> (i.e. <span><math><mrow><mi>sp</mi></mrow><mo>(</mo><mi>ϕ</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>)</mo><mo>⊆</mo><mrow><mi>sp</mi></mrow><mo>(</mo><mi>X</mi><mo>)</mo></math></span> for all <span><math><mi>X</mi><mo>∈</mo><msubsup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>≤</mo><mi>k</mi></mrow></msubsup></math></span>) either preserves characteristic polynomials or takes only nilpotent values. Moreover, for any <em>k</em> there exists a real analytic embedding of <span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>≤</mo><mi>k</mi></mrow></msubsup></math></span> into the space of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> nilpotent matrices for all sufficiently large <em>n</em>. This phenomenon cannot occur when <em>ϕ</em> is injective and either <span><math><mi>k</mi><mo>></mo><mi>n</mi><mo>−</mo><msqrt><mrow><mi>n</mi></mrow></msqrt></math></span> or the image of <em>ϕ</em> is contained in <span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>≤</mo><mi>k</mi></mrow></msubsup></math></span>. We then establish a main result of the paper – a variant of Šemrl's preserver theorem for <span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>≤</mo><mi>k</mi></mrow></msubsup></math></span>: if <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span>, any injective continuous map <span><math><mi>ϕ</mi><mo>:</mo><msubsup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>≤</mo><mi>k</mi></mrow></msubsup><mo>→</mo><msubsup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>≤</mo><mi>k</mi></mrow></msubsup></math></span> that preserves commutativity and shrinks spectrum is of the form <span><math><mi>ϕ</mi><mo>(</mo><mo>⋅</mo><mo>)</mo><mo>=</mo><mi>T</mi><mo>(</mo><mo>⋅</mo><mo>)</mo><msup><mrow><mi>T</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> or <span><math><mi>ϕ</mi><mo>(</mo><mo>⋅</mo><mo>)</mo><mo>=</mo><mi>T</mi><msup><mrow><mo>(</mo><mo>⋅</mo><mo>)</mo></mrow><mrow><mi>t</mi></mrow></msup><msup><mrow><mi>T</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"724 ","pages":"Pages 298-319"},"PeriodicalIF":1.0,"publicationDate":"2025-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144563589","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":"Higham's approximations of polar factors of operators","authors":"Abdellatif Bourhim , Mostafa Mbekhta","doi":"10.1016/j.laa.2025.06.020","DOIUrl":"10.1016/j.laa.2025.06.020","url":null,"abstract":"<div><div>Let <em>H</em> be a complex Hilbert space and <span><math><mi>B</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> be the algebra of all bounded linear operators on <em>H</em>. For any operator <span><math><mi>A</mi><mo>∈</mo><mi>B</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span>, let <span><math><mo>|</mo><mi>A</mi><mo>|</mo><mo>:</mo><mo>=</mo><msup><mrow><mo>(</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mi>A</mi><mo>)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></math></span> and <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>†</mo></mrow></msup></math></span> be the modulus and the Moore-Penrose inverse of <em>A</em>, respectively. The polar factor of an operator <span><math><mi>A</mi><mo>∈</mo><mi>B</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> is, by the polar decomposition theorem, the unique partial isometry <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>∈</mo><mi>B</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> whose kernel coincides with that of <em>A</em> such that <span><math><mi>A</mi><mo>=</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>|</mo><mi>A</mi><mo>|</mo></math></span>. In this paper, we show that if <span><math><mi>A</mi><mo>∈</mo><mi>B</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> has closed range then the sequence<span><span><span><math><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mi>A</mi><mo>,</mo><mspace></mspace><mspace></mspace><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mrow><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>+</mo><msubsup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>⁎</mo><mo>†</mo></mrow></msubsup><mo>)</mo></mrow><mo>,</mo><mspace></mspace><mo>(</mo><mi>n</mi><mo>≥</mo><mn>0</mn><mo>)</mo><mo>,</mo></math></span></span></span> converges to <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>A</mi></mrow></msub></math></span> in the norm topology of <span><math><mi>B</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span>. This extends Higham's matrix result to the setting of operators with closed range. Furthermore, we obtain another approximation of the polar factors of operators in <span><math><mi>B</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> and discuss a related linear preserving problem.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"724 ","pages":"Pages 280-297"},"PeriodicalIF":1.0,"publicationDate":"2025-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144534586","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":"Sign patterns of semimonotone matrices","authors":"Aritra Narayan Hisabia , Manideepa Saha","doi":"10.1016/j.laa.2025.06.019","DOIUrl":"10.1016/j.laa.2025.06.019","url":null,"abstract":"<div><div>A real matrix <em>A</em> is called a (strictly) semimonotone matrix if for any nonnegative nonzero vector <em>x</em>, there exists an index <em>k</em> such that <span><math><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> is positive and <span><math><msub><mrow><mo>(</mo><mi>A</mi><mi>x</mi><mo>)</mo></mrow><mrow><mi>k</mi></mrow></msub></math></span> is (positive) nonnegative. A sign pattern matrix (or sign pattern) <em>S</em> is a matrix with entries from the set <span><math><mo>{</mo><mo>+</mo><mo>,</mo><mo>−</mo><mo>,</mo><mn>0</mn><mo>}</mo></math></span>. The paper aims to study the sign pattern <em>S</em> that allows (strictly) semimonotonicity (if there exists a (strictly) semimonotone matrix with sign pattern <em>S</em>) or requires (strictly) semimonotonicity (every matrix of sign pattern <em>S</em> is (strictly) semimonotone), and to discuss a few algebraic properties of semimonotone matrices. In particular, we provide characterizations of sign pattern matrices that allow strictly semimonotonicity. We also present a necessary and sufficient condition for a sign pattern to require semimonotonicity and strictly semimonotonicity. Furthermore, we show the existence of a basis of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msup></math></span>, consisting of semimonotone and strictly semimonotone matrices. At last, we characterize rank one semimonotone and strictly semimonotone matrices.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"724 ","pages":"Pages 206-217"},"PeriodicalIF":1.0,"publicationDate":"2025-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144491080","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":"Every 2n-by-2n complex symplectic matrix is a product of n + 1 symplectic dilatations","authors":"Ralph John de la Cruz, William Nierop","doi":"10.1016/j.laa.2025.06.018","DOIUrl":"10.1016/j.laa.2025.06.018","url":null,"abstract":"<div><div>A <span><math><mn>2</mn><mi>n</mi><mo>×</mo><mn>2</mn><mi>n</mi></math></span> complex matrix <em>A</em> is symplectic if <span><math><msup><mrow><mi>A</mi></mrow><mrow><mi>T</mi></mrow></msup><mi>J</mi><mi>A</mi><mo>=</mo><mi>J</mi></math></span> where <span><math><mi>J</mi><mo>=</mo><mrow><mo>[</mo><mtable><mtr><mtd><mn>0</mn></mtd><mtd><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub></mtd></mtr><mtr><mtd><mo>−</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mo>]</mo></mrow></math></span>. We say that <em>A</em> is a <em>symplectic dilatation</em> if <em>A</em> is symplectic and is similar to <span><math><mo>[</mo><mi>a</mi><mo>]</mo><mo>⊕</mo><mo>[</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>]</mo><mo>⊕</mo><msub><mrow><mi>I</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub></math></span>. If <span><math><mi>n</mi><mo>></mo><mn>1</mn></math></span>, we show that every <span><math><mn>2</mn><mi>n</mi><mo>×</mo><mn>2</mn><mi>n</mi></math></span> complex symplectic matrix <em>A</em> is a product of <em>n</em> symplectic dilatations except when <em>A</em> is similar to <span><math><msub><mrow><mi>J</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo><mo>⊕</mo><mo>−</mo><msub><mrow><mi>I</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub></math></span>, in which case <em>A</em> is a product of <span><math><mi>n</mi><mo>+</mo><mn>1</mn></math></span> symplectic dilatations.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"724 ","pages":"Pages 242-256"},"PeriodicalIF":1.0,"publicationDate":"2025-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144514137","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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