Pub Date : 2024-11-28DOI: 10.1016/j.laa.2024.11.021
Pedro Massey
We develop a novel convergence analysis of the classical deterministic block Krylov methods for the approximation of h-dimensional dominant subspaces and low-rank approximations of matrices (where or in the case that there is no singular gap at the index h i.e., if (where denote the singular values of A, and ). Indeed, starting with a (deterministic) matrix with satisfying a compatibility assumption with some h-dimensional right dominant subspace of A, we show that block Krylov methods produce arbitrarily good approximations for both problems mentioned above. Our approach is based on recent work by Drineas, Ipsen, Kontopoulou and Magdon-Ismail on the approximation of structural left dominant subspaces. The main difference between our work and previous work on this topic is that instead of exploiting a singular gap at the prescribed index h (which is zero in this case) we exploit the nearest existing singular gaps. We include a section with numerical examples that test the performance of our main results.
{"title":"Dominant subspace and low-rank approximations from block Krylov subspaces without a prescribed gap","authors":"Pedro Massey","doi":"10.1016/j.laa.2024.11.021","DOIUrl":"10.1016/j.laa.2024.11.021","url":null,"abstract":"<div><div>We develop a novel convergence analysis of the classical deterministic block Krylov methods for the approximation of <em>h</em>-dimensional dominant subspaces and low-rank approximations of matrices <span><math><mi>A</mi><mo>∈</mo><msup><mrow><mi>K</mi></mrow><mrow><mi>m</mi><mo>×</mo><mi>n</mi></mrow></msup></math></span> (where <span><math><mi>K</mi><mo>=</mo><mi>R</mi></math></span> or <span><math><mi>C</mi><mo>)</mo></math></span> in the case that there is no singular gap at the index <em>h</em> i.e., if <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mi>h</mi></mrow></msub><mo>=</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>h</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> (where <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≥</mo><mo>…</mo><mo>≥</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>≥</mo><mn>0</mn></math></span> denote the singular values of <em>A</em>, and <span><math><mi>p</mi><mo>=</mo><mi>min</mi><mo></mo><mo>{</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>}</mo></math></span>). Indeed, starting with a (deterministic) matrix <span><math><mi>X</mi><mo>∈</mo><msup><mrow><mi>K</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>r</mi></mrow></msup></math></span> with <span><math><mi>r</mi><mo>≥</mo><mi>h</mi></math></span> satisfying a compatibility assumption with some <em>h</em>-dimensional right dominant subspace of <em>A</em>, we show that block Krylov methods produce arbitrarily good approximations for both problems mentioned above. Our approach is based on recent work by Drineas, Ipsen, Kontopoulou and Magdon-Ismail on the approximation of structural left dominant subspaces. The main difference between our work and previous work on this topic is that instead of exploiting a singular gap at the prescribed index <em>h</em> (which is zero in this case) we exploit the nearest existing singular gaps. We include a section with numerical examples that test the performance of our main results.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 112-149"},"PeriodicalIF":1.0,"publicationDate":"2024-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164619","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 : 2024-11-26DOI: 10.1016/j.laa.2024.11.017
Peter Šepitka, Roman Šimon Hilscher
In this paper we derive new existence results for conjoined bases of singular linear Hamiltonian differential systems with given qualitative (Sturmian) properties. In particular, we examine the existence of conjoined bases with invertible upper block and with prescribed number of focal points at the endpoints of the considered unbounded interval. Such results are vital for the theory of Riccati differential equations and its applications in optimal control problems. As the main tools we use a new general characterization of conjoined bases belonging to a given equivalence class (genus) and the theory of comparative index of two Lagrangian planes. We also utilize extensively the methods of matrix analysis. The results are new even for identically normal linear Hamiltonian systems. The results are also new for linear Hamiltonian systems on a compact interval, where they provide additional equivalent conditions to the classical Reid roundabout theorem about disconjugacy.
{"title":"New existence results for conjoined bases of singular linear Hamiltonian systems with given Sturmian properties","authors":"Peter Šepitka, Roman Šimon Hilscher","doi":"10.1016/j.laa.2024.11.017","DOIUrl":"10.1016/j.laa.2024.11.017","url":null,"abstract":"<div><div>In this paper we derive new existence results for conjoined bases of singular linear Hamiltonian differential systems with given qualitative (Sturmian) properties. In particular, we examine the existence of conjoined bases with invertible upper block and with prescribed number of focal points at the endpoints of the considered unbounded interval. Such results are vital for the theory of Riccati differential equations and its applications in optimal control problems. As the main tools we use a new general characterization of conjoined bases belonging to a given equivalence class (genus) and the theory of comparative index of two Lagrangian planes. We also utilize extensively the methods of matrix analysis. The results are new even for identically normal linear Hamiltonian systems. The results are also new for linear Hamiltonian systems on a compact interval, where they provide additional equivalent conditions to the classical Reid roundabout theorem about disconjugacy.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"707 ","pages":"Pages 187-224"},"PeriodicalIF":1.0,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142748261","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}
Pub Date : 2024-11-26DOI: 10.1016/j.laa.2024.11.022
Dimas José Gonçalves , Mateus Eduardo Salomão
Consider the Jordan algebra of upper triangular matrices of order two, over a field of characteristic different from two, with the Jordan product induced by the usual associative product. For every nontrivial group grading on such algebra, we describe the set of all its graded polynomial identities. Moreover, we describe a linear basis for the corresponding relatively free graded algebra.
{"title":"Graded polynomial identities for the Jordan algebra of 2 × 2 upper triangular matrices","authors":"Dimas José Gonçalves , Mateus Eduardo Salomão","doi":"10.1016/j.laa.2024.11.022","DOIUrl":"10.1016/j.laa.2024.11.022","url":null,"abstract":"<div><div>Consider the Jordan algebra of upper triangular matrices of order two, over a field of characteristic different from two, with the Jordan product induced by the usual associative product. For every nontrivial group grading on such algebra, we describe the set of all its graded polynomial identities. Moreover, we describe a linear basis for the corresponding relatively free graded algebra.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 61-92"},"PeriodicalIF":1.0,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143165268","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 : 2024-11-26DOI: 10.1016/j.laa.2024.11.023
Zhongping Ji , Genqiang Liu , Yueqiang Zhao
Simple quasi-Whittaker modules over the Schrödinger algebra of -dimensional space-time were originally introduced and classified by Cai, Cheng, Shen in their work [7]. In the present paper, our focus lies in the study of the category of quasi-Whittaker modules over . We show that each non-singular block is equivalent to the category of finite-dimensional modules over the polynomial algebra in one variable. In particular, we can give explicit realizations of simple quasi-Whittaker modules using differential operators.
{"title":"The category of quasi-Whittaker modules over the Schrödinger algebra","authors":"Zhongping Ji , Genqiang Liu , Yueqiang Zhao","doi":"10.1016/j.laa.2024.11.023","DOIUrl":"10.1016/j.laa.2024.11.023","url":null,"abstract":"<div><div>Simple quasi-Whittaker modules over the Schrödinger algebra <span><math><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> of <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-dimensional space-time were originally introduced and classified by Cai, Cheng, Shen in their work <span><span>[7]</span></span>. In the present paper, our focus lies in the study of the category of quasi-Whittaker modules over <span><math><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>. We show that each non-singular block is equivalent to the category of finite-dimensional modules over the polynomial algebra in one variable. In particular, we can give explicit realizations of simple quasi-Whittaker modules using differential operators.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 1-11"},"PeriodicalIF":1.0,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143165265","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 : 2024-11-22DOI: 10.1016/j.laa.2024.11.020
Clemens Heuberger, Jutta Rath, Roswitha Rissner
The sequence of associated primes of powers of a monomial ideal I in a polynomial ring R eventually stabilizes by a known result by Markus Brodmann. Lê Tuân Hoa gives an upper bound for the index where the stabilization occurs. This bound depends on the generators of the ideal and is obtained by separately bounding the powers of I after which said sequence is non-decreasing and non-increasing, respectively. In this paper, we focus on the latter and call the smallest such number the copersistence index. We take up the proof idea of Lê Tuân Hoa, who exploits a certain system of inequalities whose solution sets store information about the associated primes of powers of I. However, these proofs are entangled with a specific choice for the system of inequalities. In contrast to that, we present a generic ansatz to obtain an upper bound for the copersistence index that is uncoupled from this choice of the system. We establish properties for a system of inequalities to be eligible for this approach to work. We construct two suitable inequality systems to demonstrate how this ansatz yields upper bounds for the copersistence index and compare them with Hoa's. One of the two systems leads to an improvement of the bound by an exponential factor.
根据马库斯-布罗德曼（Markus Brodmann）的已知结果，多项式环 R 中单项式理想 I 的幂的相关素数序列 (Ass(R/In))n∈N 最终会趋于稳定。Lê Tuân Hoa 给出了发生稳定化的指数上限。这个上界取决于理想的生成器，是通过分别对 I 的幂级数进行上界而得到的，在 I 的幂级数之后，所述序列分别为非递减序列和非递增序列。在本文中，我们重点讨论后者，并将这样的最小数称为共存指数。我们采用了 Lê Tuân Hoa 的证明思路，他利用了某个不等式系统，该系统的解集存储了 I 的幂的相关素数的信息。与此相反，我们提出了一个通用的解析式，以获得与系统选择无关的共存指数上界。我们建立了不等式系统的属性，使这一方法能够发挥作用。我们构建了两个合适的不等式系统，以证明这种解析如何得到共存指数的上界，并将它们与 Hoa 的上界进行比较。这两个不等式系统中，有一个不等式系统的上限提高了指数倍。
{"title":"Stabilization of associated prime ideals of monomial ideals – Bounding the copersistence index","authors":"Clemens Heuberger, Jutta Rath, Roswitha Rissner","doi":"10.1016/j.laa.2024.11.020","DOIUrl":"10.1016/j.laa.2024.11.020","url":null,"abstract":"<div><div>The sequence <span><math><msub><mrow><mo>(</mo><mi>Ass</mi><mo>(</mo><mi>R</mi><mo>/</mo><msup><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo><mo>)</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span> of associated primes of powers of a monomial ideal <em>I</em> in a polynomial ring <em>R</em> eventually stabilizes by a known result by Markus Brodmann. Lê Tuân Hoa gives an upper bound for the index where the stabilization occurs. This bound depends on the generators of the ideal and is obtained by separately bounding the powers of <em>I</em> after which said sequence is non-decreasing and non-increasing, respectively. In this paper, we focus on the latter and call the smallest such number the copersistence index. We take up the proof idea of Lê Tuân Hoa, who exploits a certain system of inequalities whose solution sets store information about the associated primes of powers of <em>I</em>. However, these proofs are entangled with a specific choice for the system of inequalities. In contrast to that, we present a generic ansatz to obtain an upper bound for the copersistence index that is uncoupled from this choice of the system. We establish properties for a system of inequalities to be eligible for this approach to work. We construct two suitable inequality systems to demonstrate how this ansatz yields upper bounds for the copersistence index and compare them with Hoa's. One of the two systems leads to an improvement of the bound by an exponential factor.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"707 ","pages":"Pages 162-186"},"PeriodicalIF":1.0,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722539","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}
Pub Date : 2024-11-22DOI: 10.1016/j.laa.2024.11.019
Yasemin Alp , E. Gokcen Kocer
The almost-Riordan arrays and their inverses are investigating by the generating functions of the row sum, the alternating row sum, and the weighted row sum. The A, Z, and ω-sequences of the almost-Riordan arrays are characterized by the generating functions of these row sums. Additionally, using the generating functions of these row sums, the product of two almost-Riordan arrays is obtained.
{"title":"Characterization of almost-Riordan arrays with row sums","authors":"Yasemin Alp , E. Gokcen Kocer","doi":"10.1016/j.laa.2024.11.019","DOIUrl":"10.1016/j.laa.2024.11.019","url":null,"abstract":"<div><div>The almost-Riordan arrays and their inverses are investigating by the generating functions of the row sum, the alternating row sum, and the weighted row sum. The <em>A</em>, <em>Z</em>, and <em>ω</em>-sequences of the almost-Riordan arrays are characterized by the generating functions of these row sums. Additionally, using the generating functions of these row sums, the product of two almost-Riordan arrays is obtained.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"706 ","pages":"Pages 101-123"},"PeriodicalIF":1.0,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722518","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 : 2024-11-22DOI: 10.1016/j.laa.2024.11.018
Adrián Andrada, Agustín Garrone
A compact symplectic manifold is said to satisfy the hard-Lefschetz condition if it is possible to develop an analogue of Hodge theory for . This loosely means that there is a notion of harmonicity of differential forms in M, depending on ω alone, such that every de Rham cohomology class in has a ω-harmonic representative. In this article, we study two non-equivalent families of diagonal almost-abelian Lie algebras that admit a distinguished almost-Kähler structure and compute their cohomology explicitly. We show that they satisfy the hard-Lefschetz condition with respect to any left-invariant symplectic structure by exploiting an unforeseen connection with Kneser graphs. We also show that for some choice of parameters their associated simply connected, completely solvable Lie groups admit lattices, thereby constructing examples of almost-Kähler solvmanifolds satisfying the hard-Lefschetz condition, in such a way that their de Rham cohomology is fully known.
如果可以为（M,ω）建立霍奇理论的类似模型，那么紧凑交错流形（M,ω）就可以说满足硬-勒夫谢茨条件。这大致意味着 M 中的微分形式有一个谐波性概念，它只取决于 ω，这样 M 中的每个 de Rham 同调类都有一个 ω 谐波代表。在这篇文章中，我们研究了两个非等价的对角近阿贝尔李代数族，它们承认一个杰出的近凯勒结构，并明确地计算了它们的同调。我们利用与 Kneser 图之间未曾预料到的联系，证明它们在任何左不变交映结构方面都满足硬-Lefschetz 条件。我们还证明，在某些参数选择下，它们相关的简单相连、完全可解的李群包含晶格，从而构造出满足硬-勒菲切茨条件的近凯勒溶点的例子，这样它们的德拉姆同调就完全可知了。
{"title":"Construction of symplectic solvmanifolds satisfying the hard-Lefschetz condition","authors":"Adrián Andrada, Agustín Garrone","doi":"10.1016/j.laa.2024.11.018","DOIUrl":"10.1016/j.laa.2024.11.018","url":null,"abstract":"<div><div>A compact symplectic manifold <span><math><mo>(</mo><mi>M</mi><mo>,</mo><mi>ω</mi><mo>)</mo></math></span> is said to satisfy the hard-Lefschetz condition if it is possible to develop an analogue of Hodge theory for <span><math><mo>(</mo><mi>M</mi><mo>,</mo><mi>ω</mi><mo>)</mo></math></span>. This loosely means that there is a notion of harmonicity of differential forms in <em>M</em>, depending on <em>ω</em> alone, such that every de Rham cohomology class in has a <em>ω</em>-harmonic representative. In this article, we study two non-equivalent families of diagonal almost-abelian Lie algebras that admit a distinguished almost-Kähler structure and compute their cohomology explicitly. We show that they satisfy the hard-Lefschetz condition with respect to any left-invariant symplectic structure by exploiting an unforeseen connection with Kneser graphs. We also show that for some choice of parameters their associated simply connected, completely solvable Lie groups admit lattices, thereby constructing examples of almost-Kähler solvmanifolds satisfying the hard-Lefschetz condition, in such a way that their de Rham cohomology is fully known.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"706 ","pages":"Pages 70-100"},"PeriodicalIF":1.0,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722517","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 : 2024-11-19DOI: 10.1016/j.laa.2024.11.016
J. Guerrero
In this work, we define the Laplacian and Normalized Laplacian energies of vertices in a graph, we derive some of its properties and relate them to combinatorial, spectral and geometric quantities of the graph.
{"title":"Laplacian energies of vertices","authors":"J. Guerrero","doi":"10.1016/j.laa.2024.11.016","DOIUrl":"10.1016/j.laa.2024.11.016","url":null,"abstract":"<div><div>In this work, we define the Laplacian and Normalized Laplacian energies of vertices in a graph, we derive some of its properties and relate them to combinatorial, spectral and geometric quantities of the graph.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"706 ","pages":"Pages 124-143"},"PeriodicalIF":1.0,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722519","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 : 2024-11-19DOI: 10.1016/j.laa.2024.11.011
Sarah Klanderman , MurphyKate Montee , Andrzej Piotrowski , Alex Rice , Bryan Shader
The Seidel matrix of a tournament on n players is an skew-symmetric matrix with entries in that encapsulates the outcomes of the games in the given tournament. It is known that the determinant of an Seidel matrix is 0 if n is odd, and is an odd perfect square if n is even. This leads to the study of the set, , of square roots of determinants of Seidel matrices. It is shown that is a proper subset of for every positive even integer, and every odd integer in the interval is in for n even. The expected value and variance of over the Seidel matrices chosen uniformly at random is determined, and upper bounds on are given, and related to the Hadamard conjecture. Finally, it is shown that for infinitely many n, contains a gap (that is, there are odd integers such that but ) and several properties of the characteristic polynomials of Seidel matrices are established.
{"title":"Determinants of Seidel tournament matrices","authors":"Sarah Klanderman , MurphyKate Montee , Andrzej Piotrowski , Alex Rice , Bryan Shader","doi":"10.1016/j.laa.2024.11.011","DOIUrl":"10.1016/j.laa.2024.11.011","url":null,"abstract":"<div><div>The Seidel matrix of a tournament on <em>n</em> players is an <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> skew-symmetric matrix with entries in <span><math><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>−</mo><mn>1</mn><mo>}</mo></math></span> that encapsulates the outcomes of the games in the given tournament. It is known that the determinant of an <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> Seidel matrix is 0 if <em>n</em> is odd, and is an odd perfect square if <em>n</em> is even. This leads to the study of the set, <span><math><mi>D</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span>, of square roots of determinants of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> Seidel matrices. It is shown that <span><math><mi>D</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> is a proper subset of <span><math><mi>D</mi><mo>(</mo><mi>n</mi><mo>+</mo><mn>2</mn><mo>)</mo></math></span> for every positive even integer, and every odd integer in the interval <span><math><mo>[</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>2</mn><mo>]</mo></math></span> is in <span><math><mi>D</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> for <em>n</em> even. The expected value and variance of <span><math><mi>det</mi><mo></mo><mi>S</mi></math></span> over the <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> Seidel matrices chosen uniformly at random is determined, and upper bounds on <span><math><mi>max</mi><mo></mo><mi>D</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> are given, and related to the Hadamard conjecture. Finally, it is shown that for infinitely many <em>n</em>, <span><math><mi>D</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> contains a gap (that is, there are odd integers <span><math><mi>k</mi><mo><</mo><mi>ℓ</mi><mo><</mo><mi>m</mi></math></span> such that <span><math><mi>k</mi><mo>,</mo><mi>m</mi><mo>∈</mo><mi>D</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> but <span><math><mi>ℓ</mi><mo>∉</mo><mi>D</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span>) and several properties of the characteristic polynomials of Seidel matrices are established.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"707 ","pages":"Pages 126-151"},"PeriodicalIF":1.0,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722543","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 : 2024-11-19DOI: 10.1016/j.laa.2024.11.015
Tejbir Lohan , Chandan Maity
We consider the adjoint action of the symplectic Lie group on its Lie algebra . An element is called -real if for some . Moreover, if for some involution , then is called strongly -real. In this paper, we prove that for every element , there exists a skew-involution such that . Furthermore, we classify the strongly -real elements in . We also classify skew-Hamiltonian matrices that are similar to their negatives via a symplectic involution.
{"title":"Strongly real adjoint orbits of complex symplectic Lie group","authors":"Tejbir Lohan , Chandan Maity","doi":"10.1016/j.laa.2024.11.015","DOIUrl":"10.1016/j.laa.2024.11.015","url":null,"abstract":"<div><div>We consider the adjoint action of the symplectic Lie group <span><math><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span> on its Lie algebra <span><math><mrow><mi>sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span>. An element <span><math><mi>X</mi><mo>∈</mo><mrow><mi>sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span> is called <span><math><msub><mrow><mi>Ad</mi></mrow><mrow><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></mrow></msub></math></span>-real if <span><math><mo>−</mo><mi>X</mi><mo>=</mo><mrow><mi>Ad</mi></mrow><mo>(</mo><mi>g</mi><mo>)</mo><mi>X</mi></math></span> for some <span><math><mi>g</mi><mo>∈</mo><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span>. Moreover, if <span><math><mo>−</mo><mi>X</mi><mo>=</mo><mrow><mi>Ad</mi></mrow><mo>(</mo><mi>h</mi><mo>)</mo><mi>X</mi></math></span> for some involution <span><math><mi>h</mi><mo>∈</mo><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span>, then <span><math><mi>X</mi><mo>∈</mo><mrow><mi>sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span> is called strongly <span><math><msub><mrow><mi>Ad</mi></mrow><mrow><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></mrow></msub></math></span>-real. In this paper, we prove that for every element <span><math><mi>X</mi><mo>∈</mo><mrow><mi>sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span>, there exists a skew-involution <span><math><mi>g</mi><mo>∈</mo><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span> such that <span><math><mo>−</mo><mi>X</mi><mo>=</mo><mrow><mi>Ad</mi></mrow><mo>(</mo><mi>g</mi><mo>)</mo><mi>X</mi></math></span>. Furthermore, we classify the strongly <span><math><msub><mrow><mi>Ad</mi></mrow><mrow><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></mrow></msub></math></span>-real elements in <span><math><mrow><mi>sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span>. We also classify skew-Hamiltonian matrices that are similar to their negatives via a symplectic involution.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"706 ","pages":"Pages 144-156"},"PeriodicalIF":1.0,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142746666","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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