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Improved bounds for randomized Schatten norm estimation of numerically low-rank matrices 数值低秩矩阵的随机夏顿规范估计的改进边界
IF 1 3区 数学
Linear Algebra and its Applications Pub Date : 2025-04-08 DOI: 10.1016/j.laa.2025.04.001
Ya-Chi Chu, Alice Cortinovis
{"title":"Improved bounds for randomized Schatten norm estimation of numerically low-rank matrices","authors":"Ya-Chi Chu,&nbsp;Alice Cortinovis","doi":"10.1016/j.laa.2025.04.001","DOIUrl":"10.1016/j.laa.2025.04.001","url":null,"abstract":"<div><div>In this work, we analyze the variance of a stochastic estimator for computing Schatten norms of matrices. The estimator extracts information from a single sketch of the matrix, that is, the product of the matrix with a few standard Gaussian random vectors. While this estimator has been proposed and used in the literature before, the existing variance bounds are often pessimistic. Our work provides a new upper bound and estimates of the variance of this estimator. These theoretical findings are supported by numerical experiments, demonstrating that the new bounds are significantly tighter than the existing ones in the case of numerically low-rank matrices.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"717 ","pages":"Pages 68-93"},"PeriodicalIF":1.0,"publicationDate":"2025-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143816586","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}
引用次数: 0
Employing star complements in search for graphs with fixed rank 用星补搜索定秩图
IF 1 3区 数学
Linear Algebra and its Applications Pub Date : 2025-04-03 DOI: 10.1016/j.laa.2025.03.021
Zoran Stanić
{"title":"Employing star complements in search for graphs with fixed rank","authors":"Zoran Stanić","doi":"10.1016/j.laa.2025.03.021","DOIUrl":"10.1016/j.laa.2025.03.021","url":null,"abstract":"<div><div>A star complement in a graph <em>G</em> of order <em>n</em> is an induced subgraph <em>H</em> of order <em>t</em>, such that <em>μ</em> is an eigenvalue of multiplicity <span><math><mi>n</mi><mo>−</mo><mi>t</mi></math></span> of <em>G</em>, but not an eigenvalue of <em>H</em>. We use an idea of Torgašev to develop an algorithm based on star complements to characterize graphs with given rank (i.e., the number of non-zero eigenvalues of the adjacency matrix) or given number of eigenvalues distinct from −1. As a demonstration, we re-prove some known results concerning graphs with a comparatively small rank. By the same method, we characterize graphs having at most 6 eigenvalues distinct from −1. Comparisons with existing results are provided.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"718 ","pages":"Pages 14-29"},"PeriodicalIF":1.0,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143817518","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}
引用次数: 0
Representation of zeros of a copositive matrix via maximal cliques of a graph 用图的极大团表示合成矩阵的零
IF 1 3区 数学
Linear Algebra and its Applications Pub Date : 2025-04-01 DOI: 10.1016/j.laa.2025.03.020
Kostyukova O.I. , Tchemisova T.V.
{"title":"Representation of zeros of a copositive matrix via maximal cliques of a graph","authors":"Kostyukova O.I. ,&nbsp;Tchemisova T.V.","doi":"10.1016/j.laa.2025.03.020","DOIUrl":"10.1016/j.laa.2025.03.020","url":null,"abstract":"<div><div>There is a strong connection between copositive matrices and graph theory. Copositive matrices provide a powerful tool for formulating and approximating various challenging graph-related problems. In return, graph theory offers a rich set of concepts and techniques that can be used to explore important properties of copositive matrices, such as their eigenvalues and spectra.</div><div>This paper presents new insights into this interplay. Specifically, we focus on the set of all zeros of a copositive matrix, examining its properties and demonstrating that it can be expressed as a union of convex hulls of certain subsets of minimal zeros. We further show that these subsets are closely linked to the maximal cliques of a special graph, constructed based on the minimal zeros of the matrix.</div><div>An algorithm is explicitly described for constructing the complete set of normalized zeros and minimal zeros of a copositive matrix.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"717 ","pages":"Pages 40-55"},"PeriodicalIF":1.0,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143777425","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}
引用次数: 0
Graphs with three and four distinct eigenvalues based on circulants 基于循环的具有三个和四个不同特征值的图
IF 1 3区 数学
Linear Algebra and its Applications Pub Date : 2025-03-27 DOI: 10.1016/j.laa.2025.03.014
Milan Bašić
{"title":"Graphs with three and four distinct eigenvalues based on circulants","authors":"Milan Bašić","doi":"10.1016/j.laa.2025.03.014","DOIUrl":"10.1016/j.laa.2025.03.014","url":null,"abstract":"<div><div>In this paper, we aim to address the open questions raised in various recent papers regarding characterization of circulant graphs with three or four distinct eigenvalues in their spectra. Our focus is on providing characterizations and constructing classes of graphs falling under this specific category. We present a characterization of circulant graphs with prime number order and unitary Cayley graphs with arbitrary order, both of which possess spectra displaying three or four distinct eigenvalues. Various constructions of circulant graphs with composite orders are provided whose spectra consist of four distinct eigenvalues. These constructions primarily utilize specific subgraphs of circulant graphs that already possess two or three eigenvalues in their spectra, employing graph operations like the tensor product, the union, and the complement. Finally, we characterize the iterated line graphs of unitary Cayley graphs whose spectra contain three or four distinct eigenvalues, and we show their non-circulant nature.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"718 ","pages":"Pages 30-57"},"PeriodicalIF":1.0,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143825799","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}
引用次数: 0
A new generalization of Fielder's lemma with applications 菲尔德引理的新推广及其应用
IF 1 3区 数学
Linear Algebra and its Applications Pub Date : 2025-03-27 DOI: 10.1016/j.laa.2025.03.019
Komal Kumari, Pratima Panigrahi
{"title":"A new generalization of Fielder's lemma with applications","authors":"Komal Kumari,&nbsp;Pratima Panigrahi","doi":"10.1016/j.laa.2025.03.019","DOIUrl":"10.1016/j.laa.2025.03.019","url":null,"abstract":"<div><div>Very recently, Ma and Wu (2024) obtained a generalization of Fielder's lemma and applied it to find the adjacency, Laplacian, and signless Laplacian spectra of the <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>-product of commuting graphs. In this paper, we give a generalization of Fielder's lemma which not only generalizes the result of Ma and Wu (2024) but also enables one to find several kind of spectra of <em>H</em>-product of graphs, when <em>H</em> is an arbitrary graph. Moreover, we compute the adjacency spectrum of <em>H</em>-product of commuting graphs and the universal adjacency spectrum of <em>H</em>-product of commuting regular graphs.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"717 ","pages":"Pages 26-39"},"PeriodicalIF":1.0,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143777424","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}
引用次数: 0
The factor width rank of a matrix 矩阵的因子宽度秩
IF 1 3区 数学
Linear Algebra and its Applications Pub Date : 2025-03-27 DOI: 10.1016/j.laa.2025.03.016
Nathaniel Johnston , Shirin Moein , Sarah Plosker
{"title":"The factor width rank of a matrix","authors":"Nathaniel Johnston ,&nbsp;Shirin Moein ,&nbsp;Sarah Plosker","doi":"10.1016/j.laa.2025.03.016","DOIUrl":"10.1016/j.laa.2025.03.016","url":null,"abstract":"<div><div>A matrix is said to have factor width at most <em>k</em> if it can be written as a sum of positive semidefinite matrices that are non-zero only in a single <span><math><mi>k</mi><mo>×</mo><mi>k</mi></math></span> principal submatrix. We explore the “factor-width-<em>k</em> rank” of a matrix, which is the minimum number of rank-1 matrices that can be used in such a factor-width-at-most-<em>k</em> decomposition. We show that the factor width rank of a banded or arrowhead matrix equals its usual rank, but for other matrices they can differ. We also establish several bounds on the factor width rank of a matrix, including a tight connection between factor-width-<em>k</em> rank and the <em>k</em>-clique covering number of a graph, and we discuss how the factor width and factor width rank change when taking Hadamard products and Hadamard powers.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"716 ","pages":"Pages 32-59"},"PeriodicalIF":1.0,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143760902","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}
引用次数: 0
Asymptotic analysis of generalized orthogonal flows 广义正交流的渐近分析
IF 1 3区 数学
Linear Algebra and its Applications Pub Date : 2025-03-27 DOI: 10.1016/j.laa.2025.03.018
Yueh-Cheng Kuo , Huey-Er Lin , Shih-Feng Shieh
{"title":"Asymptotic analysis of generalized orthogonal flows","authors":"Yueh-Cheng Kuo ,&nbsp;Huey-Er Lin ,&nbsp;Shih-Feng Shieh","doi":"10.1016/j.laa.2025.03.018","DOIUrl":"10.1016/j.laa.2025.03.018","url":null,"abstract":"<div><div>In this paper, we examine a matrix differential equation that approximates the <em>k</em>-dimensional dominant eigenspace of a matrix. We determine that its solution is orthonormal, and thus we denote this solution as the generalized orthogonal flow. We also ensure its existence and uniqueness for all time <span><math><mi>t</mi><mo>∈</mo><mi>R</mi></math></span>. In addition, we construct a particular generalized orthogonal flow that possesses minimal variation. Our findings show that the path with minimal variation is identical to an Oja-like flow. Furthermore, we conduct an in-depth analysis of the asymptotic behavior and the rate of convergence of Oja-like flow.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"717 ","pages":"Pages 1-25"},"PeriodicalIF":1.0,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143769250","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}
引用次数: 0
Minimal compact operators, subdifferential of the maximum eigenvalue and semi-definite programming 最小紧算子，最大特征值的次微分和半定规划
IF 1 3区 数学
Linear Algebra and its Applications Pub Date : 2025-03-25 DOI: 10.1016/j.laa.2025.03.017
Tamara Bottazzi , Alejandro Varela
{"title":"Minimal compact operators, subdifferential of the maximum eigenvalue and semi-definite programming","authors":"Tamara Bottazzi ,&nbsp;Alejandro Varela","doi":"10.1016/j.laa.2025.03.017","DOIUrl":"10.1016/j.laa.2025.03.017","url":null,"abstract":"<div><div>We formulate the issue of minimality of self-adjoint operators on a complex Hilbert space as a semi-definite problem, linking the work by Overton in <span><span>[18]</span></span> to the characterization of minimal hermitian matrices. This motivates us to investigate the relationship between minimal self-adjoint operators and the subdifferential of the maximum eigenvalue, initially for matrices and subsequently for compact operators. In order to do it we obtain new formulas of subdifferentials of maximum eigenvalues of compact operators that become useful in these optimization problems.</div><div>Additionally, we provide formulas for the minimizing diagonals of rank one self-adjoint operators, a result that might be applied for numerical large-scale eigenvalue optimization.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"716 ","pages":"Pages 1-31"},"PeriodicalIF":1.0,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143746826","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}
引用次数: 0
On some conjectures concerning the Böttcher-Wenzel inequality for weighted Frobenius norms 关于加权Frobenius范数Böttcher-Wenzel不等式的若干猜想
IF 1 3区 数学
Linear Algebra and its Applications Pub Date : 2025-03-24 DOI: 10.1016/j.laa.2025.03.015
Wenbo Fang, Che-Man Cheng
{"title":"On some conjectures concerning the Böttcher-Wenzel inequality for weighted Frobenius norms","authors":"Wenbo Fang,&nbsp;Che-Man Cheng","doi":"10.1016/j.laa.2025.03.015","DOIUrl":"10.1016/j.laa.2025.03.015","url":null,"abstract":"<div><div>Let <em>ω</em> be a positive definite matrix. The <em>ω</em>-weighted Frobenius norm <span><math><msub><mrow><mo>‖</mo><mo>⋅</mo><mo>‖</mo></mrow><mrow><mi>ω</mi></mrow></msub></math></span> is defined by <span><math><msub><mrow><mo>‖</mo><mi>X</mi><mo>‖</mo></mrow><mrow><mi>ω</mi></mrow></msub><mo>=</mo><msqrt><mrow><mrow><mi>tr</mi></mrow><mspace></mspace><mi>ω</mi><msup><mrow><mi>X</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mi>X</mi></mrow></msqrt></math></span>. Recently, A. Mayumi, G. Kimura, H. Ohno, and D. Chruściński raised some conjectures concerning the generalized Böttcher-Wenzel inequality:<span><span><span><math><msub><mrow><mo>‖</mo><mi>X</mi><mi>Y</mi><mo>−</mo><mi>Y</mi><mi>X</mi><mo>‖</mo></mrow><mrow><mn>1</mn></mrow></msub><mo>≤</mo><mi>C</mi><msub><mrow><mo>‖</mo><mi>X</mi><mo>‖</mo></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mo>‖</mo><mi>Y</mi><mo>‖</mo></mrow><mrow><mn>3</mn></mrow></msub><mspace></mspace><mspace></mspace><mtext> for all </mtext><mi>n</mi><mo>×</mo><mi>n</mi><mtext> complex matrices </mtext><mi>X</mi><mtext> and </mtext><mi>Y</mi><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mo>‖</mo><mo>⋅</mo><mo>‖</mo></mrow><mrow><mi>i</mi></mrow></msub></math></span> (<span><math><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></math></span>) is the Frobenius norm or <em>ω</em>-weighted Frobenius norm. In this paper, the conjectures are proved when <em>X</em> and <em>Y</em> are rank one matrices.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"718 ","pages":"Pages 1-13"},"PeriodicalIF":1.0,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143808418","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}
引用次数: 0
On bounded ratios of minors of totally positive matrices 关于全正矩阵的子矩阵的有界比
IF 1 3区 数学
Linear Algebra and its Applications Pub Date : 2025-03-24 DOI: 10.1016/j.laa.2025.03.013
Daniel Soskin , Michael Gekhtman
{"title":"On bounded ratios of minors of totally positive matrices","authors":"Daniel Soskin ,&nbsp;Michael Gekhtman","doi":"10.1016/j.laa.2025.03.013","DOIUrl":"10.1016/j.laa.2025.03.013","url":null,"abstract":"<div><div>We construct several examples of bounded Laurent monomials in minors of an <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> totally positive matrix which can not be factored into a product of so called primitive bounded ratios. This disproves the conjecture about factorization of bounded ratios due to Fallat, Gekhtman, and Johnson. However, all of the found examples still satisfy the subtraction-free property also conjectured in their same work. In addition, we show that the set of all of the bounded ratios forms a polyhedral cone of dimension <span><math><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mn>2</mn><mi>n</mi></mrow></mtd></mtr><mtr><mtd><mi>n</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>−</mo><mn>2</mn><mi>n</mi></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"715 ","pages":"Pages 46-67"},"PeriodicalIF":1.0,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143714537","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}
引用次数: 0
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