{"title":"Improved error bounds for the deflated multi-preconditioned CG method","authors":"Reinhard Nabben, Julian Schramm","doi":"10.1016/j.laa.2025.02.022","DOIUrl":"10.1016/j.laa.2025.02.022","url":null,"abstract":"<div><div>Preconditioning and deflation are well-known techniques to speed up the convergence of the CG method. The concept of multiple-preconditioning however is introduced in the last decade. Recently, in <span><span>[21]</span></span>, a new adaptive preconditioned CG method is established that combines all these techniques. The main tool of the adaptive method is a new error bound for the deflated preconditioned CG method. Using this bound it is decided in each iteration if the deflated preconditioned CG method is sufficient in reducing the error or whether an acceleration by performing iterations of the multi-preconditioned CG method is needed. Here we improve this error bound. This new bound contributes to the theory of deflation methods but can also lead to new decision rules for the adaptive multi-preconditioned CG method.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"712 ","pages":"Pages 29-48"},"PeriodicalIF":1.0,"publicationDate":"2025-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143511966","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":"State transfer and readout times for trees of diameter 4","authors":"Stephen Kirkland , Christopher M. van Bommel","doi":"10.1016/j.laa.2025.02.021","DOIUrl":"10.1016/j.laa.2025.02.021","url":null,"abstract":"<div><div>We consider the state transfer properties of continuous time quantum walks on trees of diameter 4. We characterize all pairs of strongly cospectral vertices in trees of diameter 4, finding that they fall into pairs of three different types. For each type, we construct an infinite family of diameter 4 trees for which there is pretty good state transfer between the pair of strongly cospectral vertices. Moreover, for two of those types, for each tree in the infinite family, we give an explicit sequence of readout times at which the fidelity of state transfer converges to 1. For strongly cospectral vertices of the remaining type, we identify a sequence of trees and explicit readout times so that the fidelity of state transfer between the strongly cospectral vertices approaches 1.</div><div>We also prove a result of independent interest: for a graph with the property that the fidelity of state transfer between a pair of vertices at time <span><math><msub><mrow><mi>t</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> converges to 1 as <span><math><mi>k</mi><mo>→</mo><mo>∞</mo></math></span>, then the derivative of the fidelity at <span><math><msub><mrow><mi>t</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> converges to 0 as <span><math><mi>k</mi><mo>→</mo><mo>∞</mo></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"712 ","pages":"Pages 9-28"},"PeriodicalIF":1.0,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143511965","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":"Number of characteristic polynomials of matrices with bounded height","authors":"László Mérai , Igor E. Shparlinski","doi":"10.1016/j.laa.2025.02.012","DOIUrl":"10.1016/j.laa.2025.02.012","url":null,"abstract":"<div><div>We consider the set <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>Z</mi><mo>;</mo><mi>H</mi><mo>)</mo></mrow></math></span> of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span>-matrices with integer elements of size at most <em>H</em> and obtain upper and lower bounds on the number of distinct irreducible characteristic polynomials which correspond to these matrices and thus on the number of distinct eigenvalues of these matrices. In particular, we improve some results of A. Abrams, Z. Landau, J. Pommersheim and N. Srivastava (2022).</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"712 ","pages":"Pages 1-8"},"PeriodicalIF":1.0,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143508739","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 Picard groups and Jacobians of directed graphs","authors":"Jaiung Jun , Youngsu Kim , Matthew Pisano","doi":"10.1016/j.laa.2025.02.020","DOIUrl":"10.1016/j.laa.2025.02.020","url":null,"abstract":"<div><div>The Picard group of an undirected graph is a finitely generated abelian group, and the Jacobian is the torsion subgroup of the Picard group. These groups can be computed by using the Smith normal form of the Laplacian matrix of the graph or by using chip-firing games associated with the graph. One may consider its generalization to directed graphs based on the Laplacian matrix. We compute Picard groups and Jacobians for several classes of directed trees, cycles, wheel, and multipartite graphs.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"711 ","pages":"Pages 180-211"},"PeriodicalIF":1.0,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143474627","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":"Invariant subspace perturbation of a matrix with Jordan blocks","authors":"Hongguo Xu","doi":"10.1016/j.laa.2025.02.019","DOIUrl":"10.1016/j.laa.2025.02.019","url":null,"abstract":"<div><div>We investigate how invariant subspaces corresponding to a single eigenvalue will change when a matrix is perturbed. We focus on the invariant subspaces corresponding to an eigenvalue associated with the Jordan blocks that have the same size. An invariant subspace can be expressed as the range of a full column matrix. We characterize the perturbed invariant subspaces with such matrices expressed in a sum form that exhibits the fractional orders. We also provide the formulas for the coefficient matrices associated with the zero and first fractional orders. The results extend the standard invariant subspace perturbation theory.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"711 ","pages":"Pages 143-179"},"PeriodicalIF":1.0,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143445961","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}
Ansam I. Al-Aqtash , Lon H. Mitchell , Sivaram K. Narayan
{"title":"Minimum semidefinite rank of signed graphs and partial 3-trees","authors":"Ansam I. Al-Aqtash , Lon H. Mitchell , 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":"<div><div>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 <em>A</em>, it is shown that<span><span><span><math><msub><mrow><mo>‖</mo><mi>A</mi><mo>‖</mo></mrow><mrow><mi>p</mi></mrow></msub><mo>≤</mo><msup><mrow><mo>(</mo><mrow><mi>rank</mi></mrow><mspace></mspace><mi>A</mi><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn><mi>p</mi></mrow></msup><msub><mrow><mo>‖</mo><mi>A</mi><mo>‖</mo></mrow><mrow><mn>2</mn><mi>p</mi></mrow></msub><mo>≤</mo><msup><mrow><mo>(</mo><mrow><mi>rank</mi></mrow><mspace></mspace><mi>A</mi><mo>)</mo></mrow><mrow><mo>(</mo><mn>2</mn><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>2</mn><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msup><msub><mrow><mo>‖</mo><mi>A</mi><mo>‖</mo></mrow><mrow><mn>2</mn><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub><mo>,</mo><mspace></mspace><mrow><mtext>for all </mtext><mi>p</mi><mo>≥</mo><mn>1</mn></mrow></math></span></span></span> where <span><math><msub><mrow><mo>‖</mo><mo>⋅</mo><mo>‖</mo></mrow><mrow><mi>p</mi></mrow></msub></math></span> is the Schatten <em>p</em>-norm. If <span><math><mo>{</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo><mo>}</mo></math></span> is a listing of all non-zero eigenvalues (with multiplicity) of a compact operator <em>A</em>, then we show that<span><span><span><math><mrow><munder><mo>∑</mo><mrow><mi>n</mi></mrow></munder><msup><mrow><mo>|</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo><mo>|</mo></mrow><mrow><mi>p</mi></mrow></msup><mo>≤</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msubsup><mrow><mo>‖</mo><mi>A</mi><mo>‖</mo></mrow><mrow><mi>p</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msubsup><mrow><mo>‖</mo><msup><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>‖</mo></mrow><mrow><mi>p</mi><mo>/</mo><mn>2</mn></mrow><mrow><mi>p</mi><mo>/</mo><mn>2</mn></mrow></msubsup><mo>,</mo><mspace></mspace><mrow><mtext>for all </mtext><mi>p</mi><mo>≥</mo><mn>2</mn></mrow></mrow></math></span></span></span> which improves the classical Weyl's inequality <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mo>|</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo><mo>|</mo></mrow><mrow><mi>p</mi></mrow></msup><mo>≤</mo><msubsup><mrow><mo>‖</mo><mi>A</mi><mo>‖</mo></mrow><mrow><mi>p</mi></mrow><mrow><mi>p</mi></mrow></msubsup></math></span> [Proc. Nat. Acad. Sci. USA 1949]. For an <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrix <em>A</em>, we show that the function <span><math><mi>p</mi><mo>→</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</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 , Saïd Benayadi , 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 , Fuad Kittaneh , 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 , 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}
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