Pub Date : 2026-04-01Epub Date: 2026-01-06DOI: 10.1016/j.laa.2026.01.001
Richard A. Brualdi
{"title":"From the Editor-in-Chief","authors":"Richard A. Brualdi","doi":"10.1016/j.laa.2026.01.001","DOIUrl":"10.1016/j.laa.2026.01.001","url":null,"abstract":"","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"734 ","pages":"Pages 116-117"},"PeriodicalIF":1.1,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928389","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 : 2026-04-01Epub 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":"2026-04-01","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 : 2026-04-01Epub Date: 2026-01-06DOI: 10.1016/j.laa.2026.01.002
Fernando De Terán , Bruno Iannazzo
We provide a characterization for a periodic system of generalized Sylvester and conjugate-Sylvester equations, with at most one generalized conjugate-Sylvester equation, to have a unique solution when all coefficient matrices are square and all unknown matrices of the system have the same size. We also present a procedure to reduce an arbitrary system of generalized Sylvester and conjugate-Sylvester equations to periodic systems having at most one generalized conjugate-Sylvester equation. Therefore, the obtained characterization for the uniqueness of solution of periodic systems provides a characterization for general systems of generalized Sylvester and conjugate-Sylvester equations.
{"title":"Systems of standard and conjugate Sylvester equations: a characterization for the uniqueness of solution","authors":"Fernando De Terán , Bruno Iannazzo","doi":"10.1016/j.laa.2026.01.002","DOIUrl":"10.1016/j.laa.2026.01.002","url":null,"abstract":"<div><div>We provide a characterization for a periodic system of generalized Sylvester and conjugate-Sylvester equations, with at most one generalized conjugate-Sylvester equation, to have a unique solution when all coefficient matrices are square and all unknown matrices of the system have the same size. We also present a procedure to reduce an arbitrary system of generalized Sylvester and conjugate-Sylvester equations to periodic systems having at most one generalized conjugate-Sylvester equation. Therefore, the obtained characterization for the uniqueness of solution of periodic systems provides a characterization for general systems of generalized Sylvester and conjugate-Sylvester equations.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"734 ","pages":"Pages 176-192"},"PeriodicalIF":1.1,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145979683","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 : 2026-04-01Epub 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":"2026-04-01","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 : 2026-04-01Epub Date: 2026-01-09DOI: 10.1016/j.laa.2026.01.006
Hassane Benbouziane, Kaddour Chadli, Mustapha Ech-chérif El Kettani
Let be the algebra of all bounded linear operators on an infinite-dimensional complex Banach space . For a fixed integer , an operator is called k-potent operator if . In this paper, we provide a complete description of all surjective and weakly continuous maps such that is k-potent operator if and only if is k-potent operator, for any and . We also give the result in the setting of complex Hilbert spaces without the hypothesis of continuity.
{"title":"Maps on B(X) preserving k-potent operators","authors":"Hassane Benbouziane, Kaddour Chadli, Mustapha Ech-chérif El Kettani","doi":"10.1016/j.laa.2026.01.006","DOIUrl":"10.1016/j.laa.2026.01.006","url":null,"abstract":"<div><div>Let <span><math><mi>B</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> be the algebra of all bounded linear operators on an infinite-dimensional complex Banach space <span><math><mi>X</mi></math></span>. For a fixed integer <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span>, an operator <span><math><mi>A</mi><mo>∈</mo><mi>B</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is called <em>k</em>-potent operator if <span><math><msup><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>=</mo><mi>A</mi></math></span>. In this paper, we provide a complete description of all surjective and weakly continuous maps <span><math><mi>Φ</mi><mo>:</mo><mi>B</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>→</mo><mi>B</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> such that <span><math><mi>A</mi><mo>−</mo><mi>λ</mi><mi>B</mi></math></span> is <em>k</em>-potent operator if and only if <span><math><mi>Φ</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>−</mo><mi>λ</mi><mi>Φ</mi><mo>(</mo><mi>B</mi><mo>)</mo></math></span> is <em>k</em>-potent operator, for any <span><math><mi>A</mi><mo>,</mo><mi>B</mi><mo>∈</mo><mi>B</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> and <span><math><mi>λ</mi><mo>∈</mo><mi>C</mi></math></span>. We also give the result in the setting of complex Hilbert spaces without the hypothesis of continuity.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"734 ","pages":"Pages 152-175"},"PeriodicalIF":1.1,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145979682","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 : 2026-04-01Epub 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":"2026-04-01","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 : 2026-03-15Epub 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":"2026-03-15","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 : 2026-03-15Epub Date: 2025-12-08DOI: 10.1016/j.laa.2025.12.005
Hwa-Long Gau , Chi-Kwong Li , Kuo-Zhong Wang
For an complex matrix A, we study the value , which is the maximum size of an orthonormal set such that lie on the boundary of for . We give a complete characterization of matrices A with , and determine when such a matrix has reducing subspaces. Furthermore, we characterize companion matrices and nonnegative upper triangular the Toeplitz matrices A with .
{"title":"Matrices with all diagonal entries lying on the boundary of the numerical range","authors":"Hwa-Long Gau , Chi-Kwong Li , Kuo-Zhong Wang","doi":"10.1016/j.laa.2025.12.005","DOIUrl":"10.1016/j.laa.2025.12.005","url":null,"abstract":"<div><div>For an <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> complex matrix <em>A</em>, we study the value <span><math><mi>k</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, which is the maximum size of an orthonormal set <span><math><mo>{</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></math></span> such that <span><math><msubsup><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mi>A</mi><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span> lie on the boundary of <span><math><mi>W</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> for <span><math><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi></math></span>. We give a complete characterization of matrices <em>A</em> with <span><math><mi>k</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><mi>n</mi></math></span>, and determine when such a matrix has reducing subspaces. Furthermore, we characterize companion matrices and nonnegative upper triangular the Toeplitz matrices <em>A</em> with <span><math><mi>k</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><mi>n</mi></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"733 ","pages":"Pages 1-25"},"PeriodicalIF":1.1,"publicationDate":"2026-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145711964","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 : 2026-03-15Epub Date: 2025-12-12DOI: 10.1016/j.laa.2025.12.008
Filip Jonsson Kling
Consider a standard graded artinian k-algebra B and an extension of B by a new variable, for some . We will show how maximal rank properties for powers of a general linear form on A can be determined by maximal rank properties for different powers of general linear forms on B. This is then used to study Lefschetz properties of algebras that can be obtained via such extensions. In particular, it allows for a new proof that monomial complete intersections have the strong Lefschetz property over a field of characteristic zero. Moreover, it gives a recursive formula for the determinants that show up in that case. Finally, for algebras over a field of characteristic zero, we give a classification for what properties B must have for all extensions to have the weak or the strong Lefschetz property.
{"title":"Preserving Lefschetz properties after extension of variables","authors":"Filip Jonsson Kling","doi":"10.1016/j.laa.2025.12.008","DOIUrl":"10.1016/j.laa.2025.12.008","url":null,"abstract":"<div><div>Consider a standard graded artinian <em>k</em>-algebra <em>B</em> and an extension of <em>B</em> by a new variable, <span><math><mi>A</mi><mo>=</mo><mi>B</mi><msub><mrow><mo>⊗</mo></mrow><mrow><mi>k</mi></mrow></msub><mi>k</mi><mo>[</mo><mi>x</mi><mo>]</mo><mo>/</mo><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> for some <span><math><mi>d</mi><mo>≥</mo><mn>1</mn></math></span>. We will show how maximal rank properties for powers of a general linear form on <em>A</em> can be determined by maximal rank properties for different powers of general linear forms on <em>B</em>. This is then used to study Lefschetz properties of algebras that can be obtained via such extensions. In particular, it allows for a new proof that monomial complete intersections have the strong Lefschetz property over a field of characteristic zero. Moreover, it gives a recursive formula for the determinants that show up in that case. Finally, for algebras over a field of characteristic zero, we give a classification for what properties <em>B</em> must have for all extensions <span><math><mi>B</mi><msub><mrow><mo>⊗</mo></mrow><mrow><mi>k</mi></mrow></msub><mi>k</mi><mo>[</mo><mi>x</mi><mo>]</mo><mo>/</mo><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> to have the weak or the strong Lefschetz property.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"733 ","pages":"Pages 26-60"},"PeriodicalIF":1.1,"publicationDate":"2026-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145788850","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 : 2026-03-15Epub 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":"2026-03-15","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}
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