{"title":"Acceleration and restart for the randomized Bregman-Kaczmarz method","authors":"Lionel Tondji , Ion Necoara , Dirk A. Lorenz","doi":"10.1016/j.laa.2024.07.009","DOIUrl":"10.1016/j.laa.2024.07.009","url":null,"abstract":"<div><p>Optimizing strongly convex functions subject to linear constraints is a fundamental problem with numerous applications. In this work, we propose a block (accelerated) randomized Bregman-Kaczmarz method that only uses a block of constraints in each iteration to tackle this problem. We consider a dual formulation of this problem in order to deal in an efficient way with the linear constraints. Using convex tools, we show that the corresponding dual function satisfies the Polyak-Lojasiewicz (PL) property, provided that the primal objective function is strongly convex and verifies additionally some other mild assumptions. However, adapting the existing theory on coordinate descent methods to our dual formulation can only give us sublinear convergence results in the dual space. In order to obtain convergence results in some criterion corresponding to the primal (original) problem, we transfer our algorithm to the primal space, which combined with the PL property allows us to get linear convergence rates. More specifically, we provide a theoretical analysis of the convergence of our proposed method under different assumptions on the objective and demonstrate in the numerical experiments its superior efficiency and speed up compared to existing methods for the same problem.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"699 ","pages":"Pages 508-538"},"PeriodicalIF":1.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141872772","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 splittable linear Lie algebras","authors":"Zhiguang Hu, Haichuan Bai","doi":"10.1016/j.laa.2024.07.016","DOIUrl":"10.1016/j.laa.2024.07.016","url":null,"abstract":"<div><p>A linear Lie algebra is splittable if it contains the semisimple and nilpotent parts of each element. It is early known that a solvable linear Lie algebra <span><math><mi>g</mi></math></span> is splittable if and only if <span><math><mi>g</mi><mo>=</mo><mi>a</mi><mo>+</mo><mi>n</mi></math></span>, where <span><math><mi>a</mi></math></span> is an abelian subalgebra of <span><math><mi>g</mi></math></span> composed of semisimple elements and <span><math><mi>n</mi></math></span> is the ideal of all nilpotent matrices of <span><math><mi>g</mi></math></span>. In this paper, using elementary linear algebra we give a direct proof of the theorem and related results. Besides, we determine the structure of linear Lie algebras composed of semisimple or nilpotent elements.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"700 ","pages":"Pages 26-34"},"PeriodicalIF":1.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141843398","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 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":"699 ","pages":"Pages 459-507"},"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":"700 ","pages":"Pages 50-60"},"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":"699 ","pages":"Pages 421-458"},"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":"703 ","pages":"Pages 329-368"},"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":"700 ","pages":"Pages 1-25"},"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":"699 ","pages":"Pages 539-568"},"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":"700 ","pages":"Pages 35-49"},"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}
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