Tikesh Verma , Debasisha Mishra , Michael Tsatsomeros
{"title":"Cayley transform for Toeplitz and dual matrices","authors":"Tikesh Verma , Debasisha Mishra , Michael Tsatsomeros","doi":"10.1016/j.laa.2024.10.007","DOIUrl":"10.1016/j.laa.2024.10.007","url":null,"abstract":"<div><div>Let an <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> complex matrix <em>A</em> be such that <span><math><mi>I</mi><mo>+</mo><mi>A</mi></math></span> is invertible. The Cayley transform of <em>A</em>, denoted by <span><math><mi>C</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, is defined as<span><span><span><math><mi>C</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><msup><mrow><mo>(</mo><mi>I</mi><mo>+</mo><mi>A</mi><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>I</mi><mo>−</mo><mi>A</mi><mo>)</mo><mo>.</mo></math></span></span></span> Fallat and Tsatsomeros (2002) <span><span>[5]</span></span> and Mondal et al. (2024) <span><span>[15]</span></span> studied the Cayley transform of a matrix <em>A</em> in the context of P-matrices, H-matrices, M-matrices, totally positive matrices, positive definite matrices, almost skew-Hermitian matrices, and semipositive matrices. In this paper, the investigation of the Cayley transform is continued for Toeplitz matrices, circulant matrices, unipotent matrices, and dual matrices. An expression of the Cayley transform for dual matrices is established. It is shown that the Cayley transform of a dual symmetric matrix is always a dual symmetric matrix. The Cayley transform of a dual skew-symmetric matrix is discussed. The results are illustrated with examples.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"703 ","pages":"Pages 627-644"},"PeriodicalIF":1.0,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142434518","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":"Characterizing Jordan embeddings between block upper-triangular subalgebras via preserving properties","authors":"Ilja Gogić , Tatjana Petek , Mateo Tomašević","doi":"10.1016/j.laa.2024.10.005","DOIUrl":"10.1016/j.laa.2024.10.005","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> be the algebra of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> complex matrices. We consider arbitrary subalgebras <span><math><mi>A</mi></math></span> of <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> which contain the algebra of all upper-triangular matrices (i.e. block upper-triangular subalgebras), and their Jordan embeddings. We first describe Jordan embeddings <span><math><mi>ϕ</mi><mo>:</mo><mi>A</mi><mo>→</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> as maps of the form <span><math><mi>ϕ</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><mi>T</mi><mi>X</mi><msup><mrow><mi>T</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> or <span><math><mi>ϕ</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><mi>T</mi><msup><mrow><mi>X</mi></mrow><mrow><mi>t</mi></mrow></msup><msup><mrow><mi>T</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span>, where <span><math><mi>T</mi><mo>∈</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is an invertible matrix, and then we obtain a simple criteria of when one block upper-triangular subalgebra Jordan-embeds into another (and in that case we describe the form of such embeddings). As a main result, we characterize Jordan embeddings <span><math><mi>ϕ</mi><mo>:</mo><mi>A</mi><mo>→</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> (when <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span>) as continuous injective maps which preserve commutativity and spectrum. We show by counterexamples that all these assumptions are indispensable (unless <span><math><mi>A</mi><mo>=</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> when injectivity is superfluous).</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"704 ","pages":"Pages 192-217"},"PeriodicalIF":1.0,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142533209","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":"Powers of Karpelevič arcs and their sparsest realising matrices","authors":"Priyanka Joshi , Stephen Kirkland , Helena Šmigoc","doi":"10.1016/j.laa.2024.10.001","DOIUrl":"10.1016/j.laa.2024.10.001","url":null,"abstract":"<div><div>The region in the complex plane containing the eigenvalues of all <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> stochastic matrices was described by Karpelevič in 1951, and it is since then known as the Karpelevič region. The boundary of the Karpelevič region is the union of arcs called the Karpelevič arcs. We provide a complete characterization of the Karpelevič arcs that are powers of some other Karpelevič arc. Furthermore, we find the necessary and sufficient conditions for a sparsest stochastic matrix associated with the Karpelevič arc of order <em>n</em> to be a power of another stochastic matrix.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"703 ","pages":"Pages 463-503"},"PeriodicalIF":1.0,"publicationDate":"2024-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142424205","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":"Counting gradings on Lie algebras of block-triangular matrices","authors":"Diogo Diniz , Alex Ramos Borges , Eduardo Fonsêca","doi":"10.1016/j.laa.2024.10.002","DOIUrl":"10.1016/j.laa.2024.10.002","url":null,"abstract":"<div><div>We study the number of isomorphism classes of gradings on Lie algebras of block-triangular matrices. Let <em>G</em> be a finite abelian group, for <span><math><mi>m</mi><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>s</mi></mrow></msup></math></span> we determine the number <span><math><msup><mrow><mi>E</mi></mrow><mrow><mo>(</mo><mo>−</mo><mo>)</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>,</mo><mi>m</mi><mo>)</mo></math></span> of isomorphism classes of elementary <em>G</em>-gradings on the Lie algebra <span><math><mi>U</mi><mi>T</mi><msup><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow><mrow><mo>(</mo><mo>−</mo><mo>)</mo></mrow></msup></math></span> of block-triangular matrices over an algebraically closed field of characteristic zero. We study the asymptotic growth of <span><math><msup><mrow><mi>E</mi></mrow><mrow><mo>(</mo><mo>−</mo><mo>)</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>,</mo><mi>m</mi><mo>)</mo></math></span> and as a consequence prove that the <span><math><msup><mrow><mi>E</mi></mrow><mrow><mo>(</mo><mo>−</mo><mo>)</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>,</mo><mo>⋅</mo><mo>)</mo></math></span> determines <em>G</em> up to isomorphism. We also study the asymptotic growth of the number <span><math><msup><mrow><mi>N</mi></mrow><mrow><mo>(</mo><mo>−</mo><mo>)</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>,</mo><mi>m</mi><mo>)</mo></math></span> of isomorphism classes of <em>G</em>-gradings on <span><math><mi>U</mi><mi>T</mi><msup><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow><mrow><mo>(</mo><mo>−</mo><mo>)</mo></mrow></msup></math></span> and prove that <span><math><msup><mrow><mi>N</mi></mrow><mrow><mo>(</mo><mo>−</mo><mo>)</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>,</mo><mi>m</mi><mo>)</mo><mo>)</mo><mo>∼</mo><mo>|</mo><mi>G</mi><mo>|</mo><msup><mrow><mi>E</mi></mrow><mrow><mo>(</mo><mo>−</mo><mo>)</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>,</mo><mi>m</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"703 ","pages":"Pages 504-527"},"PeriodicalIF":1.0,"publicationDate":"2024-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142424092","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 joint spectral radius is pointwise Hölder continuous","authors":"Jeremias Epperlein, Fabian Wirth","doi":"10.1016/j.laa.2024.09.016","DOIUrl":"10.1016/j.laa.2024.09.016","url":null,"abstract":"<div><div>We show that the joint spectral radius is pointwise Hölder continuous. In addition, the joint spectral radius is locally Hölder continuous for <em>ε</em>-inflations. In the two-dimensional case, local Hölder continuity holds on the matrix sets with positive joint spectral radius.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"704 ","pages":"Pages 92-122"},"PeriodicalIF":1.0,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142533205","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":"Universal coacting Hopf algebra of a finite dimensional Lie-Yamaguti algebra","authors":"Saikat Goswami , Satyendra Kumar Mishra , Goutam Mukherjee","doi":"10.1016/j.laa.2024.09.017","DOIUrl":"10.1016/j.laa.2024.09.017","url":null,"abstract":"<div><div>M. E. Sweedler first constructed a universal Hopf algebra of an algebra. It is known that the dual notions to the existing ones play a dominant role in Hopf algebra theory. Yu. I. Manin and D. Tambara introduced the dual notion of Sweedler's construction in separate works. In this paper, we construct a universal algebra for a finite-dimensional Lie-Yamaguti algebra. We demonstrate that this universal algebra possesses a bialgebra structure, leading to a universal coacting Hopf algebra for a finite-dimensional Lie-Yamaguti algebra. Additionally, we develop a representation-theoretic version of our results. As an application, we characterize the automorphism group and classify all abelian group gradings of a finite-dimensional Lie-Yamaguti algebra.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"703 ","pages":"Pages 556-583"},"PeriodicalIF":1.0,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142424013","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":"Semisimple elements and the little Weyl group of real semisimple Zm-graded Lie algebras","authors":"Willem de Graaf , Hông Vân Lê","doi":"10.1016/j.laa.2024.09.015","DOIUrl":"10.1016/j.laa.2024.09.015","url":null,"abstract":"<div><div>We consider the semisimple orbits of a Vinberg <em>θ</em>-representation. First we take the complex numbers as base field. By a case by case analysis we show a technical result stating the equality of two sets of hyperplanes, one corresponding to the restricted roots of a Cartan subspace, the other corresponding to the complex reflections in the (little) Weyl group. The semisimple orbits have representatives in a finite number of sets that correspond to reflection subgroups of the (little) Weyl group. One of the consequences of our technical result is that the elements in a fixed such set all have the same stabilizer in the acting group. Secondly we study what happens when the base field is the real numbers. We look at Cartan subspaces and show that the real Cartan subspaces can be classified by the first Galois cohomology set of the normalizer of a fixed real Cartan subspace. In the real case the orbits can be classified using Galois cohomology. However, in order for that to work we need to know which orbits have a real representative. We show a theorem that characterizes the orbits of homogeneous semisimple elements that do have such a real representative. This closely follows and generalizes a theorem from <span><span>[6]</span></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"703 ","pages":"Pages 423-445"},"PeriodicalIF":1.0,"publicationDate":"2024-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142424091","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":"Extreme values of the Fiedler vector on trees","authors":"Roy R. Lederman , Stefan Steinerberger","doi":"10.1016/j.laa.2024.09.014","DOIUrl":"10.1016/j.laa.2024.09.014","url":null,"abstract":"<div><div>Let <em>G</em> be a tree on <em>n</em> vertices and let <span><math><mi>L</mi><mo>=</mo><mi>D</mi><mo>−</mo><mi>A</mi></math></span> denote the Laplacian matrix on <em>G</em>. The second-smallest eigenvalue <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>></mo><mn>0</mn></math></span>, also known as the algebraic connectivity, as well as the associated eigenvector have been of substantial interest. We investigate the question of when the maxima and minima of an associated eigenvector are assumed at the endpoints of the longest path in <em>G</em>. Our results also apply to more general graphs that ‘behave globally’ like a tree but can exhibit more complicated local structure. The crucial new ingredient is a reproducing formula for eigenvectors of graphs.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"703 ","pages":"Pages 528-555"},"PeriodicalIF":1.0,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142424093","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":"Estimations of Karcher mean by Hadamard product","authors":"Masatoshi Fujii , Yuki Seo , Masaru Tominaga","doi":"10.1016/j.laa.2024.09.013","DOIUrl":"10.1016/j.laa.2024.09.013","url":null,"abstract":"<div><div>In this paper, we estimate the difference between the Hadamard product and the Karcher mean of <em>n</em> positive invertible operators on the Hilbert space in terms of the Specht ratio and the Kantorovich constant. Also, we improve the obtained inequalities in the case of <span><math><mi>n</mi><mo>=</mo><mn>2</mn></math></span>. Moreover, we give ratio inequalities of the operator power means by the Hadamard product.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"703 ","pages":"Pages 446-462"},"PeriodicalIF":1.0,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142424204","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":"Isomorphisms between lattices of hyperinvariant subspaces","authors":"David Mingueza , M. Eulàlia Montoro , Alicia Roca","doi":"10.1016/j.laa.2024.09.012","DOIUrl":"10.1016/j.laa.2024.09.012","url":null,"abstract":"<div><div>Given two nilpotent endomorphisms, we determine when their lattices of hyperinvariant subspaces are isomorphic. The study of the lattice of hyperinvariant subspaces can be reduced to the nilpotent case when the endomorphism has a Jordan-Chevalley decomposition; for example, it occurs if the underlying field is the field of complex numbers.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"703 ","pages":"Pages 395-422"},"PeriodicalIF":1.0,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142323640","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}