{"title":"Minimal varieties of PI-algebras with graded involution","authors":"F.S. Benanti , O.M. Di Vincenzo , A. Valenti","doi":"10.1016/j.laa.2024.07.010","DOIUrl":"10.1016/j.laa.2024.07.010","url":null,"abstract":"<div><p>Let F be an algebraically closed field of characteristic zero and <em>G</em> a cyclic group of odd prime order. We consider the class of finite dimensional <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mo>⁎</mo><mo>)</mo></math></span>-algebras, namely <em>G</em>-graded algebras endowed with graded involution ⁎, and we characterize the varieties generated by algebras of this class which are minimal with respect to the <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mo>⁎</mo><mo>)</mo></math></span>-exponent.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0024379524003008/pdfft?md5=58c6ef0e78e02e4ca980c259967a3439&pid=1-s2.0-S0024379524003008-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141873394","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":"Garland's method for token graphs","authors":"Alan Lew","doi":"10.1016/j.laa.2024.07.018","DOIUrl":"10.1016/j.laa.2024.07.018","url":null,"abstract":"<div><p>The <em>k</em>-th token graph of a graph <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span> is the graph <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> whose vertices are the <em>k</em>-subsets of <em>V</em> and whose edges are all pairs of <em>k</em>-subsets <span><math><mi>A</mi><mo>,</mo><mi>B</mi></math></span> such that the symmetric difference of <em>A</em> and <em>B</em> forms an edge in <em>G</em>. Let <span><math><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the Laplacian matrix of <em>G</em>, and <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the Laplacian matrix of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. It was shown by Dalfó, Duque, Fabila-Monroy, Fiol, Huemer, Trujillo-Negrete, and Zaragoza Martínez that for any graph <em>G</em> on <em>n</em> vertices and any <span><math><mn>0</mn><mo>≤</mo><mi>ℓ</mi><mo>≤</mo><mi>k</mi><mo>≤</mo><mrow><mo>⌊</mo><mi>n</mi><mo>/</mo><mn>2</mn><mo>⌋</mo></mrow></math></span>, the spectrum of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is contained in that of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>.</p><p>Here, we continue to study the relation between the spectrum of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and that of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. In particular, we show that, for <span><math><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mrow><mo>⌊</mo><mi>n</mi><mo>/</mo><mn>2</mn><mo>⌋</mo></mrow></math></span>, any eigenvalue <em>λ</em> of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> that is not contained in the spectrum of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> satisfies<span><span><span><math><mi>k</mi><mo>(</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>−</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>≤</mo><mi>λ</mi><mo>≤</mo><mi>k</mi><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math></span> is the second smallest eigenvalue of <span><math><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> (also known ","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0024379524003082/pdfft?md5=bbdbf5062aa82eabe502f1effacd1b30&pid=1-s2.0-S0024379524003082-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141872773","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":"Convergence of the complex block Jacobi methods under the generalized serial pivot strategies","authors":"Erna Begović Kovač , Vjeran Hari","doi":"10.1016/j.laa.2024.07.012","DOIUrl":"10.1016/j.laa.2024.07.012","url":null,"abstract":"<div><p>The paper considers the convergence of the complex block Jacobi diagonalization methods under the large set of the generalized serial pivot strategies. The global convergence of the block methods for Hermitian, normal and <em>J</em>-Hermitian matrices is proven. In order to obtain the convergence results for the block methods that solve other eigenvalue problems, such as the generalized eigenvalue problem, we consider the convergence of a general block iterative process which uses the complex block Jacobi annihilators and operators.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141872774","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":"Gradings on associative triple systems of the second kind","authors":"Alberto Daza-Garcia","doi":"10.1016/j.laa.2024.07.015","DOIUrl":"10.1016/j.laa.2024.07.015","url":null,"abstract":"<div><div>On this work we study associative triple systems of the second kind. We show that for simple triple systems the automorphism group scheme is isomorphic to the automorphism group scheme of the 3-graded associative algebra with involution constructed by Loos. This result will allow us to prove our main result which is a complete classification up to isomorphism of the gradings of structurable algebras.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141785515","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the group of linear preservers of the Gau-Wu number","authors":"A. Guterman , E. Shen , I. Spitkovsky","doi":"10.1016/j.laa.2024.07.011","DOIUrl":"10.1016/j.laa.2024.07.011","url":null,"abstract":"<div><p>The Gau-Wu number is an important matrix invariant describing the geometry of the numerical range. In this work, the group of non-singular linear preservers of the Gau-Wu number is completely characterized.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141778931","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}
Aida Abiad , Bryan A. Curtis , Mary Flagg , H. Tracy Hall , Jephian C.-H. Lin , Bryan Shader
{"title":"The inverse nullity pair problem and the strong nullity interlacing property","authors":"Aida Abiad , Bryan A. Curtis , Mary Flagg , H. Tracy Hall , Jephian C.-H. Lin , Bryan Shader","doi":"10.1016/j.laa.2024.07.014","DOIUrl":"10.1016/j.laa.2024.07.014","url":null,"abstract":"<div><p>The inverse eigenvalue problem studies the possible spectra among matrices whose off-diagonal entries have their zero-nonzero patterns described by the adjacency of a graph <em>G</em>. In this paper, we refer to the <em>i</em>-nullity pair of a matrix <em>A</em> as <span><math><mo>(</mo><mi>null</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>,</mo><mi>null</mi><mo>(</mo><mi>A</mi><mo>(</mo><mi>i</mi><mo>)</mo><mo>)</mo></math></span>, where <span><math><mi>A</mi><mo>(</mo><mi>i</mi><mo>)</mo></math></span> is the matrix obtained from <em>A</em> by removing the <em>i</em>-th row and column. The inverse <em>i</em>-nullity pair problem is considered for complete graphs, cycles, and trees. The strong nullity interlacing property is introduced, and the corresponding supergraph lemma and decontraction lemma are developed as new tools for constructing matrices with a given nullity pair.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141872808","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}
Aina Mayumi , Gen Kimura , Hiromichi Ohno , Dariusz Chruściński
{"title":"Böttcher-Wenzel inequality for weighted Frobenius norms and its application to quantum physics","authors":"Aina Mayumi , Gen Kimura , Hiromichi Ohno , Dariusz Chruściński","doi":"10.1016/j.laa.2024.07.013","DOIUrl":"10.1016/j.laa.2024.07.013","url":null,"abstract":"<div><p>By employing a weighted Frobenius norm with a positive definite matrix <em>ω</em>, we introduce natural generalizations of the famous Böttcher-Wenzel (BW) inequality. Based on the combination of the weighted Frobenius norm <figure><img></figure> and the standard Frobenius norm <figure><img></figure>, there are exactly five possible generalizations, labeled (i) through (v), for the bounds on the norms of the commutator <span><math><mo>[</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>]</mo><mo>:</mo><mo>=</mo><mi>A</mi><mi>B</mi><mo>−</mo><mi>B</mi><mi>A</mi></math></span>. In this paper, we establish the tight bounds for cases (iii) and (v), and propose conjectures regarding the tight bounds for cases (i) and (ii). Additionally, the tight bound for case (iv) is derived as a corollary of case (i). All these bounds (i)-(v) serve as generalizations of the BW inequality. The conjectured bounds for cases (i) and (ii) (and thus also (iv)) are numerically supported for matrices up to size <span><math><mi>n</mi><mo>=</mo><mn>15</mn></math></span>. Proofs are provided for <span><math><mi>n</mi><mo>=</mo><mn>2</mn></math></span> and certain special cases. Interestingly, we find applications of these bounds in quantum physics, particularly in the contexts of the uncertainty relation and open quantum dynamics.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141872807","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}
A.H. Berliner , M. Catral , M. Cavers , S. Kim , P. van den Driessche
{"title":"An inverse eigenvalue problem for structured matrices determined by graph pairs","authors":"A.H. Berliner , M. Catral , M. Cavers , S. Kim , P. van den Driessche","doi":"10.1016/j.laa.2024.07.007","DOIUrl":"10.1016/j.laa.2024.07.007","url":null,"abstract":"<div><p>Given a pair of real symmetric matrices <span><math><mi>A</mi><mo>,</mo><mi>B</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msup></math></span> with nonzero patterns determined by the edges of any pair of chosen graphs on <em>n</em> vertices, we consider an inverse eigenvalue problem for the structured matrix <span><math><mi>C</mi><mo>=</mo><mrow><mo>[</mo><mtable><mtr><mtd><mspace></mspace><mi>A</mi><mspace></mspace></mtd><mtd><mi>B</mi></mtd></mtr><mtr><mtd><mspace></mspace><mi>I</mi><mspace></mspace></mtd><mtd><mi>O</mi></mtd></mtr></mtable><mo>]</mo></mrow><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>×</mo><mn>2</mn><mi>n</mi></mrow></msup></math></span>. We conjecture that <em>C</em> can attain any spectrum that is closed under conjugation. We use a structured Jacobian method to prove this conjecture for <em>A</em> and <em>B</em> of orders at most 4 or when the graph of <em>A</em> has a Hamilton path, and prove a weaker version of this conjecture for any pair of graphs with a restriction on the multiplicities of eigenvalues of <em>C</em>.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0024379524002970/pdfft?md5=fa415a9b3c87c51014076976d43924d1&pid=1-s2.0-S0024379524002970-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141710709","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":"Bounds of nullity for complex unit gain graphs","authors":"Qian-Qian Chen , Ji-Ming Guo","doi":"10.1016/j.laa.2024.07.006","DOIUrl":"10.1016/j.laa.2024.07.006","url":null,"abstract":"<div><p>A complex unit gain graph, or <span><math><mi>T</mi></math></span>-gain graph, is a triple <span><math><mi>Φ</mi><mo>=</mo><mo>(</mo><mi>G</mi><mo>,</mo><mi>T</mi><mo>,</mo><mi>φ</mi><mo>)</mo></math></span> comprised of a simple graph <em>G</em> as the underlying graph of Φ, the set of unit complex numbers <span><math><mi>T</mi><mo>=</mo><mo>{</mo><mi>z</mi><mo>∈</mo><mi>C</mi><mo>:</mo><mo>|</mo><mi>z</mi><mo>|</mo><mo>=</mo><mn>1</mn><mo>}</mo></math></span>, and a gain function <span><math><mi>φ</mi><mo>:</mo><mover><mrow><mi>E</mi></mrow><mrow><mo>→</mo></mrow></mover><mo>→</mo><mi>T</mi></math></span> with the property that <span><math><mi>φ</mi><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo><mo>=</mo><mi>φ</mi><msup><mrow><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>j</mi><mi>i</mi></mrow></msub><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span>. A cactus graph is a connected graph in which any two cycles have at most one vertex in common.</p><p>In this paper, we firstly show that there does not exist a complex unit gain graph with nullity <span><math><mi>n</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mn>2</mn><mi>m</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mn>2</mn><mi>c</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span>, where <span><math><mi>n</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, <span><math><mi>m</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and <span><math><mi>c</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> are the order, matching number, and cyclomatic number of <em>G</em>. Next, we provide a lower bound on the nullity for connected complex unit gain graphs and an upper bound on the nullity for complex unit gain bipartite graphs. Finally, we characterize all non-singular complex unit gain bipartite cactus graphs, which generalizes a result in Wong et al. (2022) <span><span>[30]</span></span>.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141701119","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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