Pub Date : 2024-10-22DOI: 10.1016/j.laa.2024.10.016
Nathaniel Johnston , Sarah Plosker
A graph is called Laplacian integral if the eigenvalues of its Laplacian matrix are all integers. We investigate the subset of these graphs whose Laplacian is furthermore diagonalized by a matrix with entries coming from a fixed set, in particular, the sets or . Such graphs include as special cases the recently-investigated families of Hadamard-diagonalizable and weakly Hadamard-diagonalizable graphs. As a combinatorial tool to aid in our investigation, we introduce a family of vectors that we call balanced, which generalizes totally balanced partitions, regular sequences, and complete partitions. We show that balanced vectors completely characterize which graph complements and complete multipartite graphs are -diagonalizable, and we furthermore prove results on diagonalizability of the Cartesian product, disjoint union, and join of graphs. Particular attention is paid to the - and -diagonalizability of the complete graphs and complete multipartite graphs. Finally, we provide a complete list of all simple, connected graphs on nine or fewer vertices that are - or -diagonalizable.
{"title":"Laplacian {−1,0,1}- and {−1,1}-diagonalizable graphs","authors":"Nathaniel Johnston , Sarah Plosker","doi":"10.1016/j.laa.2024.10.016","DOIUrl":"10.1016/j.laa.2024.10.016","url":null,"abstract":"<div><div>A graph is called <em>Laplacian integral</em> if the eigenvalues of its Laplacian matrix are all integers. We investigate the subset of these graphs whose Laplacian is furthermore diagonalized by a matrix with entries coming from a fixed set, in particular, the sets <span><math><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span> or <span><math><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>. Such graphs include as special cases the recently-investigated families of <em>Hadamard-diagonalizable</em> and <em>weakly Hadamard-diagonalizable</em> graphs. As a combinatorial tool to aid in our investigation, we introduce a family of vectors that we call <em>balanced</em>, which generalizes totally balanced partitions, regular sequences, and complete partitions. We show that balanced vectors completely characterize which graph complements and complete multipartite graphs are <span><math><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>-diagonalizable, and we furthermore prove results on diagonalizability of the Cartesian product, disjoint union, and join of graphs. Particular attention is paid to the <span><math><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>- and <span><math><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>-diagonalizability of the complete graphs and complete multipartite graphs. Finally, we provide a complete list of all simple, connected graphs on nine or fewer vertices that are <span><math><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>- or <span><math><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>-diagonalizable.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"704 ","pages":"Pages 309-339"},"PeriodicalIF":1.0,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554411","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 : 2024-10-22DOI: 10.1016/j.laa.2024.10.015
Chiara Castello
Sidon spaces have been introduced by Bachoc, Serra and Zémor as the q-analogue of Sidon sets, classical combinatorial objects introduced by Simon Szidon. In 2018 Roth, Raviv and Tamo introduced the notion of r-Sidon spaces, as an extension of Sidon spaces, which may be seen as the q-analogue of -sets, a generalization of classical Sidon sets. Thanks to their work, the interest on Sidon spaces has increased quickly because of their connection with cyclic subspace codes they pointed out. This class of codes turned out to be of interest since they can be used in random linear network coding. In this work we focus on a particular class of them, the one-orbit cyclic subspace codes, through the investigation of some properties of Sidon spaces and r-Sidon spaces, providing some upper and lower bounds on the possible dimension of their r-span and showing explicit constructions in the case in which the upper bound is achieved. Moreover, we provide further constructions of r-Sidon spaces, arising from algebraic and combinatorial objects, and we show examples of -sets constructed by means of them.
{"title":"On generalized Sidon spaces","authors":"Chiara Castello","doi":"10.1016/j.laa.2024.10.015","DOIUrl":"10.1016/j.laa.2024.10.015","url":null,"abstract":"<div><div>Sidon spaces have been introduced by Bachoc, Serra and Zémor as the <em>q</em>-analogue of Sidon sets, classical combinatorial objects introduced by Simon Szidon. In 2018 Roth, Raviv and Tamo introduced the notion of <em>r</em>-Sidon spaces, as an extension of Sidon spaces, which may be seen as the <em>q</em>-analogue of <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span>-sets, a generalization of classical Sidon sets. Thanks to their work, the interest on Sidon spaces has increased quickly because of their connection with cyclic subspace codes they pointed out. This class of codes turned out to be of interest since they can be used in random linear network coding. In this work we focus on a particular class of them, the one-orbit cyclic subspace codes, through the investigation of some properties of Sidon spaces and <em>r</em>-Sidon spaces, providing some upper and lower bounds on the possible dimension of their <em>r-span</em> and showing explicit constructions in the case in which the upper bound is achieved. Moreover, we provide further constructions of <em>r</em>-Sidon spaces, arising from algebraic and combinatorial objects, and we show examples of <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span>-sets constructed by means of them.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"704 ","pages":"Pages 270-308"},"PeriodicalIF":1.0,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554410","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 : 2024-10-21DOI: 10.1016/j.laa.2024.10.018
Truong Huu Dung , Tran Nam Son
Let D be a division ring with center F and multiplicative group , where each element of the commutator subgroup of can be expressed as a product of at most s commutators. A known theorem of Kursov states that if D is finite-dimensional over F, then every element of the commutator subgroup of the general linear group over D can be expressed as a product of at most commutators. We show that this result remains valid when F has a sufficiently large number of elements, without requiring D to be finite-dimensional. Our approach not only improves upon recent results on matrix decompositions over division rings but also provides a look at the Engel word map for matrices over arbitrary algebras.
设 D 是一个中心为 F 的分环和乘法群 D×，其中 D× 的换元子群的每个元素都可以表示为最多 s 个换元的乘积。库尔索夫的一个已知定理指出，如果 D 是 F 上的有限维，那么 D 上一般线性群的换元子群的每个元素都可以表示为最多 s+1 个换元的乘积。我们证明，当 F 有足够多的元素时，这一结果仍然有效，而不需要 D 是有限维的。我们的方法不仅改进了最近关于除法环上矩阵分解的结果，而且还提供了对任意数组上矩阵的恩格尔词映射的研究。
{"title":"On Kursov's theorem for matrices over division rings","authors":"Truong Huu Dung , Tran Nam Son","doi":"10.1016/j.laa.2024.10.018","DOIUrl":"10.1016/j.laa.2024.10.018","url":null,"abstract":"<div><div>Let <em>D</em> be a division ring with center <em>F</em> and multiplicative group <span><math><msup><mrow><mi>D</mi></mrow><mrow><mo>×</mo></mrow></msup></math></span>, where each element of the commutator subgroup of <span><math><msup><mrow><mi>D</mi></mrow><mrow><mo>×</mo></mrow></msup></math></span> can be expressed as a product of at most <em>s</em> commutators. A known theorem of Kursov states that if <em>D</em> is finite-dimensional over <em>F</em>, then every element of the commutator subgroup of the general linear group over <em>D</em> can be expressed as a product of at most <span><math><mi>s</mi><mo>+</mo><mn>1</mn></math></span> commutators. We show that this result remains valid when <em>F</em> has a sufficiently large number of elements, without requiring <em>D</em> to be finite-dimensional. Our approach not only improves upon recent results on matrix decompositions over division rings but also provides a look at the Engel word map for matrices over arbitrary algebras.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"704 ","pages":"Pages 218-230"},"PeriodicalIF":1.0,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142533210","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 : 2024-10-21DOI: 10.1016/j.laa.2024.10.020
Amrita Mandal , Bibhas Adhikari
We propose a graph theoretic approach to determine the trace of the product of two permutation matrices through a weighted digraph representation for a pair of permutation matrices. Consequently, we derive trace-zero doubly stochastic (DS) matrices of order 5 whose k-th power is also a trace-zero DS matrix for . Then, we determine necessary conditions for the coefficients of a generic polynomial of degree 5 to be realizable as the characteristic polynomial of a trace-zero DS matrix of order 5. Finally, we approximate the eigenvalue region of trace-zero DS matrices of order 5.
{"title":"On the trace-zero doubly stochastic matrices of order 5","authors":"Amrita Mandal , Bibhas Adhikari","doi":"10.1016/j.laa.2024.10.020","DOIUrl":"10.1016/j.laa.2024.10.020","url":null,"abstract":"<div><div>We propose a graph theoretic approach to determine the trace of the product of two permutation matrices through a weighted digraph representation for a pair of permutation matrices. Consequently, we derive trace-zero doubly stochastic (DS) matrices of order 5 whose <em>k</em>-th power is also a trace-zero DS matrix for <span><math><mi>k</mi><mo>∈</mo><mo>{</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>}</mo></math></span>. Then, we determine necessary conditions for the coefficients of a generic polynomial of degree 5 to be realizable as the characteristic polynomial of a trace-zero DS matrix of order 5. Finally, we approximate the eigenvalue region of trace-zero DS matrices of order 5.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"704 ","pages":"Pages 340-360"},"PeriodicalIF":1.0,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142579121","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 : 2024-10-21DOI: 10.1016/j.laa.2024.10.019
M.A. Khrystik , A.M. Maksaev
Let be the algebra of matrices over a field and let be its generating set (as an -algebra). The length of is the smallest number k such that equals the -linear span of all products of the length at most k of matrices from . The length of , denoted by , is defined to be the maximal length of any of its generating sets. In 1984, Paz conjectured that , for any field . This conjecture has been verified only for . In this paper, we prove Paz's conjecture for , meaning that . We also prove that .
设 Mn(F) 是一个域 F 上 n×n 矩阵的代数，设 S 是它的生成集（作为一个 F 代数）。S 的长度是最小数 k，使得 Mn(F) 等于来自 S 的矩阵的所有长度至多为 k 的乘积的 F 线性跨度。Mn(F) 的长度用 l(Mn(F)) 表示，定义为其任何一个生成集的最大长度。1984 年，帕兹猜想，对于任意域 F，l(Mn(F))=2n-2。在本文中，我们证明了 n=6 时帕斯的猜想，即 l(M6(F))=10。我们还证明了 12⩽l(M7(F))⩽13。
{"title":"A proof of the Paz conjecture for 6 × 6 matrices","authors":"M.A. Khrystik , A.M. Maksaev","doi":"10.1016/j.laa.2024.10.019","DOIUrl":"10.1016/j.laa.2024.10.019","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math></span> be the algebra of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices over a field <span><math><mi>F</mi></math></span> and let <span><math><mi>S</mi></math></span> be its generating set (as an <span><math><mi>F</mi></math></span>-algebra). The length of <span><math><mi>S</mi></math></span> is the smallest number <em>k</em> such that <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math></span> equals the <span><math><mi>F</mi></math></span>-linear span of all products of the length at most <em>k</em> of matrices from <span><math><mi>S</mi></math></span>. The length of <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math></span>, denoted by <span><math><mi>l</mi><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo><mo>)</mo></math></span>, is defined to be the maximal length of any of its generating sets. In 1984, Paz conjectured that <span><math><mi>l</mi><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo><mo>)</mo><mo>=</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></math></span>, for any field <span><math><mi>F</mi></math></span>. This conjecture has been verified only for <span><math><mi>n</mi><mo>⩽</mo><mn>5</mn></math></span>. In this paper, we prove Paz's conjecture for <span><math><mi>n</mi><mo>=</mo><mn>6</mn></math></span>, meaning that <span><math><mi>l</mi><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo><mo>)</mo><mo>=</mo><mn>10</mn></math></span>. We also prove that <span><math><mn>12</mn><mo>⩽</mo><mi>l</mi><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mn>7</mn></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo><mo>)</mo><mo>⩽</mo><mn>13</mn></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"704 ","pages":"Pages 249-269"},"PeriodicalIF":1.0,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554409","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 : 2024-10-18DOI: 10.1016/j.laa.2024.10.014
Tobias Breiten, Philipp Schulze
We present a new balancing-based structure-preserving model reduction technique for linear port-Hamiltonian descriptor systems. The proposed method relies on a modification of a set of two dual generalized algebraic Riccati equations that arise in the context of linear quadratic Gaussian balanced truncation for differential algebraic systems. We derive an a priori error bound with respect to a right coprime factorization of the underlying transfer function thereby allowing for an estimate with respect to the gap metric. We further theoretically and numerically analyze the influence of the Hamiltonian and a change thereof, respectively. With regard to this change of the Hamiltonian, we provide a novel procedure that is based on a recently introduced Kalman–Yakubovich–Popov inequality for descriptor systems. Numerical examples demonstrate how the quality of reduced-order models can significantly be improved by first computing an extremal solution to this inequality.
{"title":"Structure-preserving linear quadratic Gaussian balanced truncation for port-Hamiltonian descriptor systems","authors":"Tobias Breiten, Philipp Schulze","doi":"10.1016/j.laa.2024.10.014","DOIUrl":"10.1016/j.laa.2024.10.014","url":null,"abstract":"<div><div>We present a new balancing-based structure-preserving model reduction technique for linear port-Hamiltonian descriptor systems. The proposed method relies on a modification of a set of two dual generalized algebraic Riccati equations that arise in the context of linear quadratic Gaussian balanced truncation for differential algebraic systems. We derive an a priori error bound with respect to a right coprime factorization of the underlying transfer function thereby allowing for an estimate with respect to the gap metric. We further theoretically and numerically analyze the influence of the Hamiltonian and a change thereof, respectively. With regard to this change of the Hamiltonian, we provide a novel procedure that is based on a recently introduced Kalman–Yakubovich–Popov inequality for descriptor systems. Numerical examples demonstrate how the quality of reduced-order models can significantly be improved by first computing an extremal solution to this inequality.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"704 ","pages":"Pages 146-191"},"PeriodicalIF":1.0,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142533208","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 : 2024-10-17DOI: 10.1016/j.laa.2024.10.013
Mikaël Pichot , Erik Séguin
We establish new metric characterizations for the norm (respectively, ultraweak) closure of the convex hull of a bounded set in an arbitrary -algebra (respectively, von Neumann algebra), and provide applications of these results to the majorization theory.
我们为任意 C⁎代数（分别为 von Neumann 代数）中有界集的凸环的规范（分别为超弱）闭合建立了新的度量特征，并将这些结果应用于大化理论。
{"title":"Separation theorems for bounded convex sets of bounded operators","authors":"Mikaël Pichot , Erik Séguin","doi":"10.1016/j.laa.2024.10.013","DOIUrl":"10.1016/j.laa.2024.10.013","url":null,"abstract":"<div><div>We establish new metric characterizations for the norm (respectively, ultraweak) closure of the convex hull of a bounded set in an arbitrary <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebra (respectively, von Neumann algebra), and provide applications of these results to the majorization theory.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"704 ","pages":"Pages 132-145"},"PeriodicalIF":1.0,"publicationDate":"2024-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142533207","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 : 2024-10-15DOI: 10.1016/j.laa.2024.10.012
Chaiwat Namnak , Kijti Rodtes
In this short paper, we provide some refinement inequalities on generalized matrix functions. In particular, permanent inequalities concerning doubly stochastic positive semidefinite matrices are also included.
{"title":"Generalized matrix functions and some refinement inequalities","authors":"Chaiwat Namnak , Kijti Rodtes","doi":"10.1016/j.laa.2024.10.012","DOIUrl":"10.1016/j.laa.2024.10.012","url":null,"abstract":"<div><div>In this short paper, we provide some refinement inequalities on generalized matrix functions. In particular, permanent inequalities concerning doubly stochastic positive semidefinite matrices are also included.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"704 ","pages":"Pages 123-131"},"PeriodicalIF":1.0,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142533206","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 : 2024-10-11DOI: 10.1016/j.laa.2024.10.010
Sara Angela Filippini , Lorenzo Guerrieri
Let M be a perfect module of projective dimension 3 over a Gorenstein, local or graded ring R. We denote by the minimal free resolution of M. Using the generic ring associated to the format of we define higher structure maps, according to the theory developed by Weyman in [26]. We introduce a generalization of classical linkage for R-module using the Buchsbaum–Rim complex, and study the behavior of structure maps under this Buchsbaum–Rim linkage. In particular, for certain formats we obtain criteria for these R-modules to lie in the Buchsbaum–Rim linkage class of a Buchsbaum–Rim complex of length 3.
我们用 F 表示 M 的最小自由解析。根据韦曼在[26]中提出的理论，我们使用与 F 的格式相关的泛环定义高级结构映射。我们用布赫斯鲍姆-里姆复数引入了 R 模块经典联立的广义，并研究了结构映射在布赫斯鲍姆-里姆联立下的行为。特别是，对于某些格式，我们得到了这些 R 模块位于长度为 3 的布赫斯鲍姆-里姆复数的布赫斯鲍姆-里姆联结类中的标准。
{"title":"Mapping free resolutions of length three II - Module formats","authors":"Sara Angela Filippini , Lorenzo Guerrieri","doi":"10.1016/j.laa.2024.10.010","DOIUrl":"10.1016/j.laa.2024.10.010","url":null,"abstract":"<div><div>Let <em>M</em> be a perfect module of projective dimension 3 over a Gorenstein, local or graded ring <em>R</em>. We denote by <span><math><mi>F</mi></math></span> the minimal free resolution of <em>M</em>. Using the generic ring associated to the format of <span><math><mi>F</mi></math></span> we define higher structure maps, according to the theory developed by Weyman in <span><span>[26]</span></span>. We introduce a generalization of classical linkage for <em>R</em>-module using the Buchsbaum–Rim complex, and study the behavior of structure maps under this Buchsbaum–Rim linkage. In particular, for certain formats we obtain criteria for these <em>R</em>-modules to lie in the Buchsbaum–Rim linkage class of a Buchsbaum–Rim complex of length 3.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"704 ","pages":"Pages 1-34"},"PeriodicalIF":1.0,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142444612","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 : 2024-10-10DOI: 10.1016/j.laa.2024.10.009
Changpeng Shao
In Steinerberger (2021) [23] and Shao (2023) [21], two new types of Kaczmarz algorithms, which share some similarities, for consistent linear systems were proposed. These two algorithms not only compete with many previous Kaczmarz algorithms but, more importantly, reveal some interesting new geometric properties of solutions to linear systems that are not obvious from the standard viewpoint of the Kaczmarz algorithm. In this paper, we comprehensively study these two algorithms. First, we theoretically analyse the algorithms for solving least squares, which is more common in practice. Second, we extend the two algorithms to block versions and provide their rigorous estimate on the convergence rates. Third, as a theoretical complement, we provide more results on properties of the convergence rate. All these results contribute to a more thorough understanding of these algorithms.
{"title":"Reflective block Kaczmarz algorithms for least squares","authors":"Changpeng Shao","doi":"10.1016/j.laa.2024.10.009","DOIUrl":"10.1016/j.laa.2024.10.009","url":null,"abstract":"<div><div>In Steinerberger (2021) <span><span>[23]</span></span> and Shao (2023) <span><span>[21]</span></span>, two new types of Kaczmarz algorithms, which share some similarities, for consistent linear systems were proposed. These two algorithms not only compete with many previous Kaczmarz algorithms but, more importantly, reveal some interesting new geometric properties of solutions to linear systems that are not obvious from the standard viewpoint of the Kaczmarz algorithm. In this paper, we comprehensively study these two algorithms. First, we theoretically analyse the algorithms for solving least squares, which is more common in practice. Second, we extend the two algorithms to block versions and provide their rigorous estimate on the convergence rates. Third, as a theoretical complement, we provide more results on properties of the convergence rate. All these results contribute to a more thorough understanding of these algorithms.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"703 ","pages":"Pages 584-618"},"PeriodicalIF":1.0,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142424014","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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