Linear Algebra and its Applications最新文献

{"title":"Generalized matrix functions and some refinement inequalities","authors":"Chaiwat Namnak ,&nbsp;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":null,"pages":null},"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}

{"title":"Mapping free resolutions of length three II - Module formats","authors":"Sara Angela Filippini ,&nbsp;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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"title":"Strong quantum state transfer on graphs via loop edges","authors":"Gabor Lippner,&nbsp;Yujia Shi","doi":"10.1016/j.laa.2024.10.008","DOIUrl":"10.1016/j.laa.2024.10.008","url":null,"abstract":"<div><div>We quantify the effect of weighted loops at the source and target nodes of a graph on the strength of quantum state transfer between these vertices. We give lower bounds on loop weights that guarantee strong transfer fidelity that works for any graph where this protocol is feasible. By considering local spectral symmetry, we show that the required weight size depends only on the maximum degree of the graph and, in some less favorable cases, the distance between vertices. Additionally, we explore the duration for which transfer strength remains above a specified threshold.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142533204","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":"On the cardinality of matrices with prescribed rank and partial trace over a finite field","authors":"Kumar Balasubramanian ,&nbsp;Krishna Kaipa ,&nbsp;Himanshi Khurana","doi":"10.1016/j.laa.2024.10.011","DOIUrl":"10.1016/j.laa.2024.10.011","url":null,"abstract":"<div><div>Let <em>F</em> be the finite field of order <em>q</em> and <span><math><mi>M</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> be the set of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices of rank <em>r</em> over the field <em>F</em>. For <span><math><mi>α</mi><mo>∈</mo><mi>F</mi></math></span> and <span><math><mi>A</mi><mo>∈</mo><mi>M</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span>, let<span><span><span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>r</mi></mrow><mrow><mi>α</mi></mrow></msubsup><mo>=</mo><mrow><mo>{</mo><mi>X</mi><mo>∈</mo><mi>M</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>F</mi><mo>)</mo><mo>|</mo><mi>Tr</mi><mo>(</mo><mi>A</mi><mi>X</mi><mo>)</mo><mo>=</mo><mi>α</mi><mo>}</mo></mrow><mo>.</mo></math></span></span></span> In this article, we solve the problem of determining the cardinality of <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>r</mi></mrow><mrow><mi>α</mi></mrow></msubsup></math></span>. We also solve the generalization of the problem to rectangular matrices.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142444534","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 affine spaces of rectangular matrices with constant rank","authors":"Clément de Seguins Pazzis","doi":"10.1016/j.laa.2024.10.006","DOIUrl":"10.1016/j.laa.2024.10.006","url":null,"abstract":"<div><div>Let <span><math><mi>F</mi></math></span> be a field, and <span><math><mi>n</mi><mo>≥</mo><mi>p</mi><mo>≥</mo><mi>r</mi><mo>&gt;</mo><mn>0</mn></math></span> be integers. In a recent article, Rubei has determined, when <span><math><mi>F</mi></math></span> is the field of real numbers, the greatest possible dimension for an affine subspace of <em>n</em>–by–<em>p</em> matrices with entries in <span><math><mi>F</mi></math></span> in which all the elements have rank <em>r</em>. In this note, we generalize her result to an arbitrary field with more than <span><math><mi>r</mi><mo>+</mo><mn>1</mn></math></span> elements, and we classify the spaces that reach the maximal dimension as a function of the classification of the affine subspaces of invertible matrices of <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math></span> with dimension <span><math><mo>(</mo><mtable><mtr><mtd><mi>s</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></math></span>. The latter is known to be connected to the classification of nonisotropic quadratic forms over <span><math><mi>F</mi></math></span> up to congruence.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142444535","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 note on an invariant distance of the bidisk","authors":"Deepak K. D. ,&nbsp;Kenta Kojin ,&nbsp;Michio Seto","doi":"10.1016/j.laa.2024.10.003","DOIUrl":"10.1016/j.laa.2024.10.003","url":null,"abstract":"<div><div>In this short paper, we discuss relation between an invariant distance of the bidisk and Kreĭn space geometry. In particular, an interpolation theorem for rational maps with respect to our invariant distance is proven.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142434517","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 the uniqueness and computation of commuting extensions","authors":"Pascal Koiran","doi":"10.1016/j.laa.2024.10.004","DOIUrl":"10.1016/j.laa.2024.10.004","url":null,"abstract":"<div><div>A tuple <span><math><mo>(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math></span> of matrices of size <span><math><mi>r</mi><mo>×</mo><mi>r</mi></math></span> is said to be a <em>commuting extension</em> of a tuple <span><math><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math></span> of matrices of size <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> if the <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> pairwise commute and each <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> sits in the upper left corner of a block decomposition of <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> (here, <em>r</em> and <em>n</em> are two arbitrary integers with <span><math><mi>n</mi><mo>&lt;</mo><mi>r</mi></math></span>). This notion was discovered and rediscovered in several contexts including algebraic complexity theory (in Strassen's work on tensor rank), in numerical analysis for the construction of cubature formulas and in quantum mechanics for the study of computational methods and the study of the so-called “quantum Zeno dynamics.” Commuting extensions have also attracted the attention of the linear algebra community. In this paper we present 3 types of results:<ul><li><span>(i)</span><span><div>Theorems on the uniqueness of commuting extensions for three matrices or more.</div></span></li><li><span>(ii)</span><span><div>Algorithms for the computation of commuting extensions of minimal size. These algorithms work under the same assumptions as our uniqueness theorems. They are applicable up to <span><math><mi>r</mi><mo>=</mo><mn>4</mn><mi>n</mi><mo>/</mo><mn>3</mn></math></span>, and are apparently the first provably efficient algorithms for this problem applicable beyond <span><math><mi>r</mi><mo>=</mo><mi>n</mi><mo>+</mo><mn>1</mn></math></span>.</div></span></li><li><span>(iii)</span><span><div>A genericity theorem showing that our algorithms and uniqueness theorems can be applied to a wide range of input matrices.</div></span></li></ul></div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142434519","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":"Cayley transform for Toeplitz and dual matrices","authors":"Tikesh Verma ,&nbsp;Debasisha Mishra ,&nbsp;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":null,"pages":null},"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 ,&nbsp;Tatjana Petek ,&nbsp;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":null,"pages":null},"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}