Pub Date : 2026-01-02DOI: 10.1016/j.laa.2025.12.022
Chenyue Feng, Shoumin Liu , Xumin Wang
In this paper, we will compute the characteristic polynomials for finite dimensional representations of classical complex Lie algebras and the exceptional Lie algebra of type , which can be obtained through the orbits of integral weights under the action of their corresponding Weyl groups and the invariant polynomial theory of the Weyl groups. We show that the characteristic polynomials can be decomposed into products of irreducible orbit factors, each of which is invariant under the action of their corresponding Weyl groups.
{"title":"Characteristic polynomials for classical Lie algebras and their orbit decompositions","authors":"Chenyue Feng, Shoumin Liu , Xumin Wang","doi":"10.1016/j.laa.2025.12.022","DOIUrl":"10.1016/j.laa.2025.12.022","url":null,"abstract":"<div><div>In this paper, we will compute the characteristic polynomials for finite dimensional representations of classical complex Lie algebras and the exceptional Lie algebra of type <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, which can be obtained through the orbits of integral weights under the action of their corresponding Weyl groups and the invariant polynomial theory of the Weyl groups. We show that the characteristic polynomials can be decomposed into products of irreducible orbit factors, each of which is invariant under the action of their corresponding Weyl groups.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"734 ","pages":"Pages 28-49"},"PeriodicalIF":1.1,"publicationDate":"2026-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928390","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}
Pub Date : 2025-12-31DOI: 10.1016/j.laa.2025.12.017
Wojciech Młotkowski , Marek Skrzypczyk , Michał Wojtylak
We compute the quadratic embedding constant for complete bipartite graphs with disjoint edges removed. Moreover, we study the quadratic embedding property for theta graphs, i.e., graphs consisting of three paths with common initial points and common endpoints. As a result, we provide an infinite family of primary graphs that are not quadratically embeddable.
{"title":"On quadratic embeddability of bipartite graphs and theta graphs","authors":"Wojciech Młotkowski , Marek Skrzypczyk , Michał Wojtylak","doi":"10.1016/j.laa.2025.12.017","DOIUrl":"10.1016/j.laa.2025.12.017","url":null,"abstract":"<div><div>We compute the quadratic embedding constant for complete bipartite graphs with disjoint edges removed. Moreover, we study the quadratic embedding property for theta graphs, i.e., graphs consisting of three paths with common initial points and common endpoints. As a result, we provide an infinite family of primary graphs that are not quadratically embeddable.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"734 ","pages":"Pages 89-115"},"PeriodicalIF":1.1,"publicationDate":"2025-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928393","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}
Pub Date : 2025-12-30DOI: 10.1016/j.laa.2025.12.018
Shankhadeep Mondal, Ram Narayan Mohapatra
This paper investigates the optimization of dual frame pairs in the context of erasure problems in data transmission, using a graph theoretical approach. Frames are essential for mitigating errors and signal loss due to their redundancy properties. We address the use of spectral radius and operator norm for error measurements, presenting conditions for the optimality of dual pairs for one and two erasures. Our study shows that a tight frame generated by connected graphs and its canonical dual pair is optimal for one-erasure scenarios. Additionally, we compute the spectral radius of the error operator for one and two erasures in graph-generated frames, establishing necessary conditions for dual pair optimality.
{"title":"Optimal dual frame pairs: A synergy with graph theory","authors":"Shankhadeep Mondal, Ram Narayan Mohapatra","doi":"10.1016/j.laa.2025.12.018","DOIUrl":"10.1016/j.laa.2025.12.018","url":null,"abstract":"<div><div>This paper investigates the optimization of dual frame pairs in the context of erasure problems in data transmission, using a graph theoretical approach. Frames are essential for mitigating errors and signal loss due to their redundancy properties. We address the use of spectral radius and operator norm for error measurements, presenting conditions for the optimality of dual pairs for one and two erasures. Our study shows that a tight frame generated by connected graphs and its canonical dual pair is optimal for one-erasure scenarios. Additionally, we compute the spectral radius of the error operator for one and two erasures in graph-generated frames, establishing necessary conditions for dual pair optimality.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"734 ","pages":"Pages 1-27"},"PeriodicalIF":1.1,"publicationDate":"2025-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145895934","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}
Pub Date : 2025-12-23DOI: 10.1016/j.laa.2025.12.016
Daniel Vitas
The L'vov-Kaplansky conjecture states that the image of a multilinear noncommutative polynomial f in the matrix algebra is a vector space for every . We prove this conjecture for the case where f has degree 3 and K is an algebraically closed field of characteristic 0.
{"title":"The L'vov-Kaplansky conjecture for polynomials of degree three","authors":"Daniel Vitas","doi":"10.1016/j.laa.2025.12.016","DOIUrl":"10.1016/j.laa.2025.12.016","url":null,"abstract":"<div><div>The L'vov-Kaplansky conjecture states that the image of a multilinear noncommutative polynomial <em>f</em> in the matrix algebra <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>K</mi><mo>)</mo></math></span> is a vector space for every <span><math><mi>n</mi><mo>∈</mo><mi>N</mi></math></span>. We prove this conjecture for the case where <em>f</em> has degree 3 and <em>K</em> is an algebraically closed field of characteristic 0.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"733 ","pages":"Pages 205-232"},"PeriodicalIF":1.1,"publicationDate":"2025-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145880820","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}
Pub Date : 2025-12-19DOI: 10.1016/j.laa.2025.12.014
David Aleja , Julio Flores , Eva Primo , Daniel Rodríguez , Miguel Romance
In this paper we analyze PageRank of a complex network as a function of its personalization vector. By using this approach, a complete characterization of the existence and uniqueness of fixed points of the PageRank of a graph is given in terms of the number and nature of its strongly connected components. The method presented essentially follows the classic Power's Method by means of a feedback-PageRank that allows to precisely compute the fixed points, in terms of the (left-hand) Perron vector of each strongly connected component.
{"title":"Fixed points of personalized PageRank centrality: From irreducible to reducible networks","authors":"David Aleja , Julio Flores , Eva Primo , Daniel Rodríguez , Miguel Romance","doi":"10.1016/j.laa.2025.12.014","DOIUrl":"10.1016/j.laa.2025.12.014","url":null,"abstract":"<div><div>In this paper we analyze PageRank of a complex network as a function of its personalization vector. By using this approach, a complete characterization of the existence and uniqueness of fixed points of the PageRank of a graph is given in terms of the number and nature of its strongly connected components. The method presented essentially follows the classic <em>Power's Method</em> by means of a <em>feedback-PageRank</em> that allows to precisely compute the fixed points, in terms of the (left-hand) Perron vector of each strongly connected component.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"733 ","pages":"Pages 233-272"},"PeriodicalIF":1.1,"publicationDate":"2025-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145880818","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}
Pub Date : 2025-12-17DOI: 10.1016/j.laa.2025.12.009
Beata Derȩgowska , Simon Foucart , Barbara Lewandowska
It is shown in this note that one can decide whether an n-dimensional subspace of is isometrically isomorphic to by testing a finite number of determinental inequalities. As a byproduct, an elementary proof is provided for the fact that an n-dimensional subspace of with projection constant equal to one must be isometrically isomorphic to .
{"title":"When is a subspace of ℓ∞N isometrically isomorphic to ℓ∞n?","authors":"Beata Derȩgowska , Simon Foucart , Barbara Lewandowska","doi":"10.1016/j.laa.2025.12.009","DOIUrl":"10.1016/j.laa.2025.12.009","url":null,"abstract":"<div><div>It is shown in this note that one can decide whether an <em>n</em>-dimensional subspace of <span><math><msubsup><mrow><mi>ℓ</mi></mrow><mrow><mo>∞</mo></mrow><mrow><mi>N</mi></mrow></msubsup></math></span> is isometrically isomorphic to <span><math><msubsup><mrow><mi>ℓ</mi></mrow><mrow><mo>∞</mo></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> by testing a finite number of determinental inequalities. As a byproduct, an elementary proof is provided for the fact that an <em>n</em>-dimensional subspace of <span><math><msubsup><mrow><mi>ℓ</mi></mrow><mrow><mo>∞</mo></mrow><mrow><mi>N</mi></mrow></msubsup></math></span> with projection constant equal to one must be isometrically isomorphic to <span><math><msubsup><mrow><mi>ℓ</mi></mrow><mrow><mo>∞</mo></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"733 ","pages":"Pages 171-177"},"PeriodicalIF":1.1,"publicationDate":"2025-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145788849","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}
Pub Date : 2025-12-16DOI: 10.1016/j.laa.2025.12.012
Bohui Ban , Xuzhou Zhan , Yongjian Hu
This paper presents a complete description of the set consisting of all allowable McMillan degrees of a nonzero Toeplitz matrix (not necessarily square) and an explicit formula for the rational generating functions of such matrices with a prescribed allowable McMillan degree. This analysis extends the earlier work by Heinig and Rost concerning the rational generating functions of a nonsingular Toeplitz matrix.
{"title":"On the rational generating functions of Toeplitz matrices","authors":"Bohui Ban , Xuzhou Zhan , Yongjian Hu","doi":"10.1016/j.laa.2025.12.012","DOIUrl":"10.1016/j.laa.2025.12.012","url":null,"abstract":"<div><div>This paper presents a complete description of the set consisting of all allowable McMillan degrees of a nonzero Toeplitz matrix (not necessarily square) and an explicit formula for the rational generating functions of such matrices with a prescribed allowable McMillan degree. This analysis extends the earlier work by Heinig and Rost concerning the rational generating functions of a nonsingular Toeplitz matrix.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"733 ","pages":"Pages 155-170"},"PeriodicalIF":1.1,"publicationDate":"2025-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145788848","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}
Pub Date : 2025-12-16DOI: 10.1016/j.laa.2025.12.013
Aikaterini Aretaki , Maria Adam , Michael Tsatsomeros
It is well known that the eigenvalues of a complex matrix A are located to the left of the vertical line passing through the largest eigenvalue of its Hermitian part, . Adam and Tsatsomeros in [1] defined a cubic algebraic curve, known as the shell of A, using the two largest eigenvalues of . This curve localizes the spectrum further and lies to the left of the aforementioned vertical line. Later, Bergqvist in [5] extended the methodology employed in [1] to define a new curve, , in terms of the three largest eigenvalues of . This article delves into the geometry of for a real matrix A to address some open questions raised in [5]. In particular, specific conditions are established to characterize the configurations of in certain cases. Additionally, the number of eigenvalues of A surrounded by a bounded branch of the curve is examined. Examples are used to validate our findings and demonstrate the quality of as a finer spectrum localization area when compared to .
{"title":"Curves and spectrum localization for real matrices","authors":"Aikaterini Aretaki , Maria Adam , Michael Tsatsomeros","doi":"10.1016/j.laa.2025.12.013","DOIUrl":"10.1016/j.laa.2025.12.013","url":null,"abstract":"<div><div>It is well known that the eigenvalues of a complex matrix <em>A</em> are located to the left of the vertical line passing through the largest eigenvalue of its Hermitian part, <span><math><mi>H</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span>. Adam and Tsatsomeros in <span><span>[1]</span></span> defined a cubic algebraic curve, known as the <em>shell</em> <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span> of <em>A</em>, using the two largest eigenvalues of <span><math><mi>H</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span>. This curve localizes the spectrum further and lies to the left of the aforementioned vertical line. Later, Bergqvist in <span><span>[5]</span></span> extended the methodology employed in <span><span>[1]</span></span> to define a new curve, <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, in terms of the three largest eigenvalues of <span><math><mi>H</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span>. This article delves into the geometry of <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span> for a real matrix <em>A</em> to address some open questions raised in <span><span>[5]</span></span>. In particular, specific conditions are established to characterize the configurations of <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span> in certain cases. Additionally, the number of eigenvalues of <em>A</em> surrounded by a bounded branch of the curve is examined. Examples are used to validate our findings and demonstrate the quality of <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span> as a finer spectrum localization area when compared to <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"733 ","pages":"Pages 116-154"},"PeriodicalIF":1.1,"publicationDate":"2025-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145788853","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}
Pub Date : 2025-12-15DOI: 10.1016/j.laa.2025.12.011
Oksana Bezushchak
D. Benkovič described Jordan homomorphisms of algebras of triangular matrices over a commutative unital ring without additive 2-torsion. We extend this result to the case of noncommutative rings and remove the assumption of additive torsion.
Let R be an associative unital algebra over a commutative unital ring Φ. Consider the algebra of triangular matrices over R, and its subalgebra consisting of matrices whose main diagonal entries lie in Φ. We prove that for any Jordan homomorphism of , its restriction to is standard.
{"title":"Jordan homomorphisms of triangular algebras over noncommutative algebras","authors":"Oksana Bezushchak","doi":"10.1016/j.laa.2025.12.011","DOIUrl":"10.1016/j.laa.2025.12.011","url":null,"abstract":"<div><div>D. Benkovič described Jordan homomorphisms of algebras of triangular matrices over a commutative unital ring without additive 2-torsion. We extend this result to the case of noncommutative rings and remove the assumption of additive torsion.</div><div>Let <em>R</em> be an associative unital algebra over a commutative unital ring Φ. Consider the algebra <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span> of triangular <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices over <em>R</em>, and its subalgebra <span><math><msubsup><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msubsup><mo>(</mo><mi>R</mi><mo>)</mo></math></span> consisting of matrices whose main diagonal entries lie in Φ. We prove that for any Jordan homomorphism of <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span>, its restriction to <span><math><msubsup><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msubsup><mo>(</mo><mi>R</mi><mo>)</mo></math></span> is standard.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"733 ","pages":"Pages 61-74"},"PeriodicalIF":1.1,"publicationDate":"2025-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145788851","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}
Pub Date : 2025-12-15DOI: 10.1016/j.laa.2025.12.007
Vance Faber , Jörg Liesen , Petr Tichý
We derive, similar to Lau and Riha in [22], a matrix formulation of a general best approximation theorem of Singer for the special case of spectral approximations of a given matrix from a given subspace. Using our matrix formulation we describe the relation of the spectral approximation problem to semidefinite programming, and we present a simple MATLAB code to solve the problem numerically. We then obtain geometric characterizations of spectral approximations that are based on the k-dimensional field of k matrices, which we illustrate with several numerical examples. The general spectral approximation problem is a min-max problem, whose value is bounded from below by the corresponding max-min problem. Using our geometric characterizations of spectral approximations, we derive several necessary and sufficient as well as sufficient conditions for equality of the max-min and min-max values. Finally, we prove that the max-min and min-max values are always equal for block diagonal matrices containing two identical diagonal blocks. Several results in this paper generalize results that have been obtained in the convergence analysis of the GMRES method for solving linear algebraic systems.
{"title":"Matrix best approximation in the spectral norm","authors":"Vance Faber , Jörg Liesen , Petr Tichý","doi":"10.1016/j.laa.2025.12.007","DOIUrl":"10.1016/j.laa.2025.12.007","url":null,"abstract":"<div><div>We derive, similar to Lau and Riha in <span><span>[22]</span></span>, a matrix formulation of a general best approximation theorem of Singer for the special case of spectral approximations of a given matrix from a given subspace. Using our matrix formulation we describe the relation of the spectral approximation problem to semidefinite programming, and we present a simple MATLAB code to solve the problem numerically. We then obtain geometric characterizations of spectral approximations that are based on the <em>k</em>-dimensional field of <em>k</em> matrices, which we illustrate with several numerical examples. The general spectral approximation problem is a min-max problem, whose value is bounded from below by the corresponding max-min problem. Using our geometric characterizations of spectral approximations, we derive several necessary and sufficient as well as sufficient conditions for equality of the max-min and min-max values. Finally, we prove that the max-min and min-max values are always equal for <span><math><mn>2</mn><mo>×</mo><mn>2</mn></math></span> block diagonal matrices containing two identical diagonal blocks. Several results in this paper generalize results that have been obtained in the convergence analysis of the GMRES method for solving linear algebraic systems.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"733 ","pages":"Pages 178-204"},"PeriodicalIF":1.1,"publicationDate":"2025-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145880819","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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