Linear Algebra and its Applications最新文献

{"title":"Minimum semidefinite rank of signed graphs and partial 3-trees","authors":"Ansam I. Al-Aqtash ,&nbsp;Lon H. Mitchell ,&nbsp;Sivaram K. Narayan","doi":"10.1016/j.laa.2025.02.018","DOIUrl":"10.1016/j.laa.2025.02.018","url":null,"abstract":"<div><div>In this paper, the sign patterns of real symmetric positive semidefinite matrices are used to study the real minimum semidefinite rank of signed graphs. The signed graphs whose real minimum semidefinite rank is one are characterized. It is shown that the real minimum semidefinite rank of a signed graph is at most the order of the graph minus two if and only if the signed graph contains a positive cycle. By considering orthogonal vertex removal in signed graphs it is shown that the real minimum semidefinite rank of a partial 3-tree is equal to its associated reduction number.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"711 ","pages":"Pages 126-142"},"PeriodicalIF":1.0,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143436972","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 inequalities for Hilbert space operators with applications","authors":"Pintu Bhunia","doi":"10.1016/j.laa.2025.02.015","DOIUrl":"10.1016/j.laa.2025.02.015","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Several unitarily invariant norm inequalities and numerical radius inequalities for Hilbert space operators are studied. We investigate some necessary and sufficient conditions for the parallelism of two bounded operators. For a finite rank operator &lt;em&gt;A&lt;/em&gt;, it is shown that&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;rank&lt;/mi&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;rank&lt;/mi&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mrow&gt;&lt;mtext&gt;for all &lt;/mtext&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; where &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; is the Schatten &lt;em&gt;p&lt;/em&gt;-norm. If &lt;span&gt;&lt;math&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is a listing of all non-zero eigenvalues (with multiplicity) of a compact operator &lt;em&gt;A&lt;/em&gt;, then we show that&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;munder&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mrow&gt;&lt;mtext&gt;for all &lt;/mtext&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; which improves the classical Weyl's inequality &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt; [Proc. Nat. Acad. Sci. USA 1949]. For an &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; matrix &lt;em&gt;A&lt;/em&gt;, we show that the function &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/m","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"711 ","pages":"Pages 40-67"},"PeriodicalIF":1.0,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143427955","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":"Bialgebra theory for nearly associative algebras and LR-algebras: Equivalence, characterization, and LR-Yang-Baxter equation","authors":"Elisabete Barreiro ,&nbsp;Saïd Benayadi ,&nbsp;Carla Rizzo","doi":"10.1016/j.laa.2025.02.017","DOIUrl":"10.1016/j.laa.2025.02.017","url":null,"abstract":"<div><div>We develop the bialgebra theory for two classes of non-associative algebras: nearly associative algebras and <em>LR</em>-algebras. In particular, building on recent studies that reveal connections between these algebraic structures, we establish that nearly associative bialgebras and <em>LR</em>-bialgebras are, in fact, equivalent concepts. We also provide a characterization of these bialgebra classes based on the coproduct. Moreover, since the development of nearly associative bialgebras — and by extension, <em>LR</em>-bialgebras — requires the framework of nearly associative <em>L</em>-algebras, we introduce this class of non-associative algebras and explore their fundamental properties. Furthermore, we identify and characterize a special class of nearly associative bialgebras, the coboundary nearly associative bialgebras, which provides a natural framework for studying the Yang-Baxter equation (YBE) within this context.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"711 ","pages":"Pages 84-125"},"PeriodicalIF":1.0,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143436971","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":"Improved numerical radius bounds using the Moore-Penrose inverse","authors":"Pintu Bhunia ,&nbsp;Fuad Kittaneh ,&nbsp;Satyajit Sahoo","doi":"10.1016/j.laa.2025.02.013","DOIUrl":"10.1016/j.laa.2025.02.013","url":null,"abstract":"<div><div>Using the Moore-Penrose inverse of a bounded linear operator, we obtain few bounds for the numerical radius, which improve the classical ones. Applying these improvements, we study equality conditions of the existing bounds. It is shown that if <em>T</em> is a bounded linear operator with closed range, then<span><span><span><math><msup><mrow><mi>w</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>T</mi><mo>)</mo><mo>≤</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>‖</mo><msup><mrow><mi>T</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mi>T</mi><mo>+</mo><mi>T</mi><msup><mrow><mi>T</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>‖</mo><mrow><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>‖</mo><mi>T</mi><msup><mrow><mi>T</mi></mrow><mrow><mi>†</mi></mrow></msup><mo>+</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>†</mi></mrow></msup><mi>T</mi><mo>‖</mo><mo>)</mo></mrow><mo>≤</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>‖</mo><msup><mrow><mi>T</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mi>T</mi><mo>+</mo><mi>T</mi><msup><mrow><mi>T</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>‖</mo><mo>.</mo></math></span></span></span> For a finite-dimensional space operator <em>T</em>, this improvement is proper if and only if <span><math><mi>R</mi><mi>a</mi><mi>n</mi><mi>g</mi><mi>e</mi><mo>(</mo><mi>T</mi><mo>)</mo><mspace></mspace><mo>∩</mo><mspace></mspace><mi>R</mi><mi>a</mi><mi>n</mi><mi>g</mi><mi>e</mi><mo>(</mo><msup><mrow><mi>T</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>)</mo><mo>=</mo><mo>{</mo><mn>0</mn><mo>}</mo></math></span>. Clearly, if <span><math><mo>‖</mo><mi>T</mi><msup><mrow><mi>T</mi></mrow><mrow><mi>†</mi></mrow></msup><mo>+</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>†</mi></mrow></msup><mi>T</mi><mo>‖</mo><mo>=</mo><mn>1</mn></math></span>, then <span><math><msup><mrow><mi>w</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>T</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>‖</mo><msup><mrow><mi>T</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mi>T</mi><mo>+</mo><mi>T</mi><msup><mrow><mi>T</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>‖</mo></math></span>. Among other results, we obtain inner product inequalities for the sum of operators, and as an application of these inequalities, we deduce relevant operator norm and numerical radius bounds.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"711 ","pages":"Pages 1-16"},"PeriodicalIF":1.0,"publicationDate":"2025-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143422618","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":"Products of Hermitian matrices over division rings","authors":"Peeraphat Gatephan ,&nbsp;Kijti Rodtes","doi":"10.1016/j.laa.2025.02.016","DOIUrl":"10.1016/j.laa.2025.02.016","url":null,"abstract":"<div><div>In this paper, we investigate the Dieudonné determinant of Hermitian matrices over division rings with involution. We prove that every matrix over division rings whose center contains at least <span><math><mi>n</mi><mo>+</mo><mn>2</mn></math></span> elements can be expressed as a product of three diagonalizable matrices. Moreover, we establish necessary and sufficient conditions for matrices to be factored into a product of a finite number of Hermitian matrices over division rings and one diagonalizable matrix for which its Dieudonné determinant is the commutator class containing one. As a consequence, Radjavi's factorization over complex numbers and over the real quaternion division ring can be obtained immediately.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 531-545"},"PeriodicalIF":1.0,"publicationDate":"2025-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143419392","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":"Concave functions and positive block matrices","authors":"Eun-Young Lee","doi":"10.1016/j.laa.2025.02.014","DOIUrl":"10.1016/j.laa.2025.02.014","url":null,"abstract":"<div><div>Let <span><math><mi>f</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> be a nonnegative concave function on <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span>. For a positive (semidefinite) matrix partitioned into four blocks such that <span><math><mi>A</mi><mi>X</mi><mo>=</mo><mi>X</mi><mi>A</mi></math></span> or <span><math><mi>B</mi><mi>X</mi><mo>=</mo><mi>X</mi><mi>B</mi></math></span>, we prove that<span><span><span><math><mrow><mo>‖</mo><mi>f</mi><mrow><mo>(</mo><mrow><mo>[</mo><mtable><mtr><mtd><mi>A</mi></mtd><mtd><mi>X</mi></mtd></mtr><mtr><mtd><msup><mrow><mi>X</mi></mrow><mrow><mo>⁎</mo></mrow></msup></mtd><mtd><mi>B</mi></mtd></mtr></mtable><mo>]</mo></mrow><mo>)</mo></mrow><mo>‖</mo></mrow><mo>≤</mo><mn>2</mn><mrow><mo>‖</mo><mi>f</mi><mrow><mo>(</mo><mfrac><mrow><mi>A</mi><mo>+</mo><mi>B</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow><mo>‖</mo></mrow></math></span></span></span> for all unitarily invariant norms. The case <span><math><mi>X</mi><mo>=</mo><mn>0</mn></math></span> is already new, contains two classical trace inequalities due to Rotfel'd and von Neumann, and generalizes an important basic majorization. Our proof is based, and also extends, a theorem of Bourin and Mhanna involving the width of the numerical range of <em>X</em>. For Schatten <em>q</em>-quasinorms, <span><math><mn>0</mn><mo>&lt;</mo><mi>q</mi><mo>&lt;</mo><mn>1</mn></math></span>, and nonnegative convex functions vanishing at 0, we obtain the reverse inequality.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"712 ","pages":"Pages 49-58"},"PeriodicalIF":1.0,"publicationDate":"2025-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143520918","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":"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}