Linear Algebra and its Applications最新文献

{"title":"Decomposable numerical ranges of normal matrices","authors":"Pan-Shun Lau , Chi-Kwong Li , Nung-Sing Sze","doi":"10.1016/j.laa.2025.05.016","DOIUrl":"10.1016/j.laa.2025.05.016","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub></math></span> (<span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>) be the set of <span><math><mi>n</mi><mo>×</mo><mi>k</mi></math></span> (<span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span>) complex matrices, and <span><math><mrow><mi>per</mi></mrow><mo>(</mo><mi>X</mi><mo>)</mo></math></span> be the permanent of a square matrix <em>X</em>. We study the three types of generalized numerical ranges associated with generalized matrix functions<span><span><span><math><msub><mrow><mi>Π</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><mrow><mo>{</mo><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>k</mi></mrow></munderover><msub><mrow><mo>(</mo><msup><mrow><mi>V</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mi>A</mi><mi>V</mi><mo>)</mo></mrow><mrow><mi>i</mi><mi>i</mi></mrow></msub><mo>:</mo><mi>V</mi><mo>∈</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>,</mo><mspace></mspace><msup><mrow><mi>V</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mi>V</mi><mo>=</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></mrow><mo>,</mo></math></span></span></span><span><span><span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><mrow><mo>{</mo><mi>det</mi><mo>⁡</mo><mo>(</mo><msup><mrow><mi>V</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mi>A</mi><mi>V</mi><mo>)</mo><mo>:</mo><mi>V</mi><mo>∈</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>,</mo><mspace></mspace><msup><mrow><mi>V</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mi>V</mi><mo>=</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></mrow><mo>,</mo></math></span></span></span> and<span><span><span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><mrow><mo>{</mo><mrow><mi>per</mi></mrow><mo>(</mo><msup><mrow><mi>V</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mi>A</mi><mi>V</mi><mo>)</mo><mo>:</mo><mi>V</mi><mo>∈</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>,</mo><mspace></mspace><msup><mrow><mi>V</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mi>V</mi><mo>=</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></mrow><mo>.</mo></math></span></span></span> We give complete descriptions of the set <span><math><msub><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span> for essentially hermitian matrices <span><math><mi>A</mi><mo>∈</mo><msub><mrow><mi>M</mi></mrow><mrow>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"722 ","pages":"Pages 237-254"},"PeriodicalIF":1.0,"publicationDate":"2025-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144139034","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":"Zero product preserving additive maps on triangular algebras","authors":"Hoger Ghahramani,&nbsp;Neda Ghoreishi,&nbsp;Saber Naseri","doi":"10.1016/j.laa.2025.05.017","DOIUrl":"10.1016/j.laa.2025.05.017","url":null,"abstract":"<div><div>Suppose that <span><math><mi>T</mi><mi>r</mi><mi>i</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>M</mi><mo>,</mo><mi>B</mi><mo>)</mo></math></span> is a unital triangular algebra, where <span><math><mi>M</mi></math></span> is a faithful <span><math><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo></math></span>-bimodule, and <span><math><mi>U</mi></math></span> is a unital algebra. Let <span><math><mi>θ</mi><mo>:</mo><mi>T</mi><mi>r</mi><mi>i</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>M</mi><mo>,</mo><mi>B</mi><mo>)</mo><mo>→</mo><mi>U</mi></math></span> be a bijective zero product preserving additive map. We show that under some mild conditions <em>θ</em> is a product of a central invertible element and a ring isomorphism. Our result applies to block upper triangular matrix algebras, nest algebras on Banach spaces and nest subalgebras of von Neumann algebras.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"722 ","pages":"Pages 178-189"},"PeriodicalIF":1.0,"publicationDate":"2025-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144134744","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 order isomorphisms of locally parallel sets between C⁎-algebras: The case of matrix algebras","authors":"Koki Igarashi ,&nbsp;Jumpei Nakamura ,&nbsp;Saikat Roy ,&nbsp;Ryotaro Tanaka","doi":"10.1016/j.laa.2025.05.018","DOIUrl":"10.1016/j.laa.2025.05.018","url":null,"abstract":"<div><div>We introduce the notion of locally parallel sets of elements of <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebras which generalizes that of norm attainment sets of continuous functions, and consider a preserver problem on order isomorphisms of locally parallel sets. It is shown that unitary multiplications of Jordan ⁎-isomorphisms give examples of order isomorphisms of locally parallel sets. This is achieved by analyzing the quasi-strong Birkhoff-James orthogonality, a new orthogonality relation between the original Birkhoff-James orthogonality and the strong Birkhoff-James orthogonality. Moreover, it is shown that every order isomorphism of locally parallel sets on the <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebra of all <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> complex matrices is essentially implemented by two unitary matrices provided that <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"722 ","pages":"Pages 190-219"},"PeriodicalIF":1.0,"publicationDate":"2025-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144139156","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":"Positive moments forever: Undecidable and decidable cases","authors":"Gemma De les Coves ,&nbsp;Joshua Graf ,&nbsp;Andreas Klingler ,&nbsp;Tim Netzer","doi":"10.1016/j.laa.2025.05.015","DOIUrl":"10.1016/j.laa.2025.05.015","url":null,"abstract":"<div><div>We investigate the generalized moment membership problem for matrices, a formulation equivalent to Skolem's problem for linear recurrence sequences. We show decidability for orthogonal, unitary, and real eigenvalue matrices, and undecidability for matrices over certain commutative and non-commutative polynomial rings. As consequences, we deduce that positivity is decidable for simple unitary linear recurrence sequences and undecidable for linear recurrence sequences over commutative polynomial rings. As a byproduct, we also prove a free version of Pólya's theorem.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"722 ","pages":"Pages 255-275"},"PeriodicalIF":1.0,"publicationDate":"2025-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144138998","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":"Symmetry in complex unit gain graphs and their spectra","authors":"Pepijn Wissing,&nbsp;Edwin R. van Dam","doi":"10.1016/j.laa.2025.05.012","DOIUrl":"10.1016/j.laa.2025.05.012","url":null,"abstract":"<div><div>Complex unit gain graphs may exhibit various kinds of symmetry. In this work, we explore structural symmetry, spectral symmetry and sign-symmetry in such graphs, and their respective relations to one-another. Our main result is a construction that transforms an arbitrary complex unit gain graph into infinitely many switching-distinct ones whose spectral symmetry does not imply sign-symmetry. This provides a more general answer to the analogue of an existence question that was recently treated in the context of signed graphs.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"722 ","pages":"Pages 164-177"},"PeriodicalIF":1.0,"publicationDate":"2025-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144134743","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 structure theory for regular graphs with fixed smallest eigenvalue","authors":"Qianqian Yang ,&nbsp;Jack H. Koolen","doi":"10.1016/j.laa.2025.05.013","DOIUrl":"10.1016/j.laa.2025.05.013","url":null,"abstract":"<div><div>In this paper we will give a structure theory for regular graphs with fixed smallest eigenvalue. As a consequence of this theory, we show that a <em>k</em>-regular graph with smallest eigenvalue at least −<em>λ</em> has clique number linear in <em>k</em> if <em>k</em> is large with respect to <em>λ</em>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"722 ","pages":"Pages 114-124"},"PeriodicalIF":1.0,"publicationDate":"2025-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144106481","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":"The automorphism group and plain eigenvalues of a graph","authors":"Wei Wang,&nbsp;Xinyue Wang","doi":"10.1016/j.laa.2025.05.011","DOIUrl":"10.1016/j.laa.2025.05.011","url":null,"abstract":"<div><div>We introduce the <em>plain polynomial</em> associated with a graph <em>G</em>, which is defined to be the quotient of the characteristic polynomial and the main polynomial of <em>G</em>. For a graph <em>G</em> with a square-free plain polynomial, we establish an upper bound on the order of its automorphism group in terms of the number of irreducible factors of the plain polynomial over <span><math><mi>Q</mi></math></span>. This improves the previous upper bound using the characteristic polynomial (G. Criscuolo, C.-M. Kwok, A. Mowshowitz, and R. Tortora, The group and the minimal polynomial of a graph, J. Combin. Theory Ser. B 29 (1980) 293–302).</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"722 ","pages":"Pages 154-163"},"PeriodicalIF":1.0,"publicationDate":"2025-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144134742","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":"Arbitrarily finely divisible stochastic matrices","authors":"Priyanka Joshi,&nbsp;Helena Šmigoc","doi":"10.1016/j.laa.2025.05.010","DOIUrl":"10.1016/j.laa.2025.05.010","url":null,"abstract":"<div><div>We introduce and study the class of arbitrarily finely divisible stochastic matrices (<span><math><msub><mrow><mi>AFD</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span>-matrices): stochastic matrices that have a stochastic <em>c</em>-th root for infinitely many natural numbers <em>c</em>. This notion generalises the class of embeddable stochastic matrices. In particular, if <em>A</em> is a transition matrix for a Markov process over some time period, then arbitrary finely divisibility of <em>A</em> inside the set of stochastic matrices is the necessary and sufficient condition for the existence of a transition matrix corresponding to this Markov process over infinitesimally short periods.</div><div>Our research explores the connection between the spectral properties of an <span><math><msub><mrow><mi>AFD</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span>-matrix <em>A</em> and the spectral properties of a limit point <em>L</em> of its stochastic roots. This connection, which is first formalised in the broader context of complex and real square matrices, poses restrictions on <em>A</em> assuming <em>L</em> is given. For example, if an <span><math><msub><mrow><mi>AFD</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span>-matrix <em>A</em> has a corresponding irreducible limit point <em>L</em>, then <em>A</em> has to be a circulant matrix. We identify all matrices that can be a limit point of stochastic roots for some non-singular <span><math><msub><mrow><mi>AFD</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span>-matrix. Further, we demonstrate a construction of a class of <span><math><msub><mrow><mi>AFD</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span>-matrices with a given limit point <em>L</em> from embeddable matrices. To illustrate these theoretical findings, we examine specific cases, including <span><math><mn>2</mn><mo>×</mo><mn>2</mn></math></span> matrices, <span><math><mn>3</mn><mo>×</mo><mn>3</mn></math></span> circulant matrices, and offer a complete characterisation of <span><math><msub><mrow><mi>AFD</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span>-matrices of rank-two.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"722 ","pages":"Pages 125-153"},"PeriodicalIF":1.0,"publicationDate":"2025-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144116178","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":"Optimization flow for approximating a matrix state involving orthonormal constraints","authors":"Bing-Ze Lu ,&nbsp;Matthew M. Lin ,&nbsp;Yu-Chen Shu","doi":"10.1016/j.laa.2025.05.009","DOIUrl":"10.1016/j.laa.2025.05.009","url":null,"abstract":"<div><div>In this work, we introduce a continuous-time dynamical flow. The purpose of this flow is to approximate a matrix state while precisely adhering to orthonormal constraints. Additionally, we apply restrictions on the probability distribution that expand beyond these constraints. Our work contributes in two ways. Firstly, we demonstrate in theory that our proposed flow guarantees convergence to the stationary point of the objective function. It consistently reduces the value of this function for almost any initial value. Secondly, we show that our approach can retrieve the decomposition of a given matrix. Even if the matrix is not inherently decomposable, our results illustrate that our approach remains reliable in obtaining optimal solutions.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"722 ","pages":"Pages 220-236"},"PeriodicalIF":1.0,"publicationDate":"2025-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144138997","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":"New families of trees determined by their spectra","authors":"Zhibin Du ,&nbsp;Carlos M. da Fonseca","doi":"10.1016/j.laa.2025.05.007","DOIUrl":"10.1016/j.laa.2025.05.007","url":null,"abstract":"<div><div>In a groundbreaking work, Rowlinson in 2010 established some bounds for the multiplicities of an eigenvalue of a tree. These limits were obtained using the star complement technique and have been the subject of increasing interest in recent years. In this paper, we refine them and as a consequence we obtain new families of trees determined by their spectra. For this purpose, we develop a new method based on the eigenvalue multiplicities. As special cases, we can recover the spectral characterization recently obtained for the <em>p</em>-sun and the double <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span>-sun.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"722 ","pages":"Pages 101-113"},"PeriodicalIF":1.0,"publicationDate":"2025-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144099653","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}