{"title":"The factor width rank of a matrix","authors":"Nathaniel Johnston , Shirin Moein , 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}
{"title":"Minimal compact operators, subdifferential of the maximum eigenvalue and semi-definite programming","authors":"Tamara Bottazzi , 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}
{"title":"On bounded ratios of minors of totally positive matrices","authors":"Daniel Soskin , 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}
{"title":"On the phase retrievability of irreducible representations of finite groups","authors":"Chuangxun Cheng","doi":"10.1016/j.laa.2025.03.011","DOIUrl":"10.1016/j.laa.2025.03.011","url":null,"abstract":"<div><div>Let <em>G</em> be a finite group and <span><math><mi>π</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>U</mi><mo>(</mo><mi>V</mi><mo>)</mo></math></span> be an irreducible representation of <em>G</em> on a complex Hilbert space <em>V</em>. In this paper we study the phase retrieval property of <em>π</em> and the existence of maximal spanning vectors for <span><math><mo>(</mo><mi>π</mi><mo>,</mo><mi>G</mi><mo>,</mo><mi>V</mi><mo>)</mo></math></span>. By translating the existence of maximal spanning vectors into the existence of cyclic vectors of the form <span><math><mi>v</mi><mo>⊗</mo><mi>v</mi></math></span> for the representation <span><math><mo>(</mo><mi>π</mi><mo>⊗</mo><msup><mrow><mi>π</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>,</mo><mi>G</mi><mo>,</mo><mi>V</mi><mo>⊗</mo><mi>V</mi><mo>)</mo></math></span>, we show that if <em>π</em> is unramified, in the sense that each irreducible component of <span><math><mi>π</mi><mo>⊗</mo><msup><mrow><mi>π</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> has multiplicity one, then <em>π</em> admits maximal spanning vectors and hence does phase retrieval. Moreover, if <span><math><mi>GA</mi><mo>(</mo><mn>1</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> is the one-dimensional affine group over the finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> and <span><math><mi>π</mi><mo>:</mo><mi>GA</mi><mo>(</mo><mn>1</mn><mo>,</mo><mi>q</mi><mo>)</mo><mo>→</mo><mi>U</mi><mo>(</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> is the unique <span><math><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-dimensional irreducible representation of <span><math><mi>GA</mi><mo>(</mo><mn>1</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> (which is ramified), we give a characterization of maximal spanning vectors for <span><math><mo>(</mo><mi>π</mi><mo>,</mo><mi>GA</mi><mo>(</mo><mn>1</mn><mo>,</mo><mi>q</mi><mo>)</mo><mo>,</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> by a detailed study of the adjoint representation of <span><math><mi>GA</mi><mo>(</mo><mn>1</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> on <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>GA</mi><mo>(</mo><mn>1</mn><mo>,</mo><mi>q</mi><mo>)</mo><mo>)</mo></math></span>. In particular, we show that the set of maximal spanning vectors are open dense in <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> and the representation <span><math><mo>(</mo><mi>π</mi><mo>,</mo><mi>GA</mi><mo>(</mo><mn>1</mn><mo>,</mo><mi>q</mi><mo>)</mo><mo>,</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> does phase retrieval. Furthermore, we show that the special representations and the cuspidal representations of <span><math><msub><mrow><mi>G","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"714 ","pages":"Pages 64-95"},"PeriodicalIF":1.0,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143697737","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":"Spectral radius and fractional [a,b]-factor of graphs","authors":"Yuang Li , Dandan Fan , Yinfen Zhu","doi":"10.1016/j.laa.2025.03.012","DOIUrl":"10.1016/j.laa.2025.03.012","url":null,"abstract":"<div><div>Let <span><math><mi>h</mi><mo>:</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>→</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span> be a function on <span><math><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and let <span><math><mi>a</mi><mo>,</mo><mi>b</mi></math></span> be two positive integers with <span><math><mi>a</mi><mo>≤</mo><mi>b</mi></math></span>. If <span><math><mi>a</mi><mo>≤</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>e</mi><mo>∈</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo></mrow></msub><mi>h</mi><mo>(</mo><mi>e</mi><mo>)</mo><mo>≤</mo><mi>b</mi></math></span> for any <span><math><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, then the spanning subgraph with edge set <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>h</mi></mrow></msub><mo>=</mo><mo>{</mo><mi>e</mi><mo>∈</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mi>h</mi><mo>(</mo><mi>e</mi><mo>)</mo><mo>></mo><mn>0</mn><mo>}</mo></math></span>, denoted by <span><math><mi>G</mi><mrow><mo>[</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>h</mi></mrow></msub><mo>]</mo></mrow></math></span>, is called a fractional <span><math><mo>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>]</mo></math></span>-factor of <em>G</em> with indicator function <em>h</em>. In this paper, we provide a spectral condition to guarantee the existence of a fractional <span><math><mo>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>]</mo></math></span>-factor in a graph with minimum degree <span><math><mi>δ</mi><mo>≥</mo><mi>a</mi><mo>≥</mo><mn>1</mn></math></span>, which extends some previous results. Moreover, we also provide a lower bound on the size of a graph to guarantee the existence of a fractional <span><math><mo>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>]</mo></math></span>-factor for <span><math><mi>b</mi><mo>≥</mo><mi>a</mi><mo>≥</mo><mn>1</mn></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"715 ","pages":"Pages 32-45"},"PeriodicalIF":1.0,"publicationDate":"2025-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143706244","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 joint numerical radius","authors":"Amit Maji , Atanu Manna , Ram Mohapatra","doi":"10.1016/j.laa.2025.03.008","DOIUrl":"10.1016/j.laa.2025.03.008","url":null,"abstract":"<div><div>We investigate the Crawford number and numerical radius of model operators on Hilbert spaces. For an <em>n</em>-tuple of doubly commuting shifts, the joint numerical radius and the joint Crawford number are determined. Additionally, we use the Hermite-Hadamard inequality and the Orlicz function to derive new and improved joint numerical radius inequalities of operators on Hilbert spaces.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"715 ","pages":"Pages 17-31"},"PeriodicalIF":1.0,"publicationDate":"2025-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143681502","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 limit points of the smallest positive eigenvalues of graphs","authors":"Sasmita Barik, Debabrota Mondal","doi":"10.1016/j.laa.2025.03.006","DOIUrl":"10.1016/j.laa.2025.03.006","url":null,"abstract":"<div><div>In 1972, Hoffman <span><span>[11]</span></span> initiated the study of limit points of eigenvalues of nonnegative symmetric integer matrices. He posed the question of finding all limit points of the set of spectral radii of all nonnegative symmetric integer matrices. In the same article, the author demonstrated that it is enough to consider the adjacency matrices of simple graphs to study the limit points of spectral radii. Since then, many researchers have worked on similar problems, considering various specific eigenvalues such as the least eigenvalue, the <em>k</em>th largest eigenvalue, and the <em>k</em>th smallest eigenvalue, among others. Motivated by this, we ask the question, “which real numbers are the limit points of the set of the smallest positive eigenvalues (respectively, the largest negative eigenvalues) of graphs?” In this article, we provide a complete answer to this question by proving that any nonnegative (respectively, nonpositive) real number is a limit point of the set of all smallest positive eigenvalues (respectively, largest negative eigenvalues) of graphs. We also show that the union of the sets of limit points of the smallest positive eigenvalues and the largest negative eigenvalues of graphs is dense in <span><math><mi>R</mi></math></span>, the set of all real numbers.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"715 ","pages":"Pages 1-16"},"PeriodicalIF":1.0,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143681501","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":"Unitary similarity and the numerical radius preservers","authors":"Abdellatif Bourhim , Mohamed Mabrouk","doi":"10.1016/j.laa.2025.03.005","DOIUrl":"10.1016/j.laa.2025.03.005","url":null,"abstract":"<div><div>Let <span><math><mi>B</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> be the algebra of all bounded linear operators acting on a separable infinite-dimensional complex Hilbert space <span><math><mi>H</mi></math></span>, and denote by <span><math><mi>w</mi><mo>(</mo><mi>T</mi><mo>)</mo></math></span> the numerical radius of any operator <span><math><mi>T</mi><mo>∈</mo><mi>B</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span>. In this paper, we describe the form of all bijective linear maps <em>ϕ</em> on <span><math><mi>B</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> for which <span><math><mi>w</mi><mo>(</mo><mi>ϕ</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>)</mo><mo>=</mo><mi>w</mi><mo>(</mo><mi>ϕ</mi><mo>(</mo><mi>S</mi><mo>)</mo><mo>)</mo></math></span> whenever <span><math><mi>T</mi><mo>,</mo><mspace></mspace><mi>S</mi><mo>∈</mo><mi>B</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> are two unitarily similar operators.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"714 ","pages":"Pages 15-27"},"PeriodicalIF":1.0,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143683388","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":"Christensen-Sinclair factorization via semidefinite programming","authors":"Francisco Escudero-Gutiérrez","doi":"10.1016/j.laa.2025.03.007","DOIUrl":"10.1016/j.laa.2025.03.007","url":null,"abstract":"<div><div>We show that the Christensen-Sinclair factorization theorem, when the underlying Hilbert spaces are finite dimensional, is an instance of strong duality of semidefinite programming. This gives an elementary proof of the result and also provides an efficient algorithm to compute the Christensen-Sinclair factorization.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"714 ","pages":"Pages 28-44"},"PeriodicalIF":1.0,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143683389","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":"Interval global optimization problem in max-plus algebra","authors":"Helena Myšková, Ján Plavka","doi":"10.1016/j.laa.2025.03.009","DOIUrl":"10.1016/j.laa.2025.03.009","url":null,"abstract":"<div><div>Consider the global optimization problem of minimizing the max-plus product <span><math><mi>A</mi><mo>⊗</mo><mi>x</mi></math></span>, where <em>A</em> is a given matrix and the constraint set is the set of column vectors <em>x</em> such that the sum of products <span><math><msub><mrow><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msub><mspace></mspace><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span> is equal to <em>c</em> and <em>c</em> is a given positive real constant, <span><math><msub><mrow><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span> are non-negative numbers with sum equal to 1. We show that the solvability of the given global optimization problem is independent of the number <em>c</em> if the components of the vector <em>x</em> can also be negative. From a practical point of view, we further consider the solvability of the global optimization problem with non-negative constraints. We propose an algorithm which decides whether a given problem is solvable, extend the problem to interval matrices and provide an algorithm to verify the solvability of interval global optimization problem.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"714 ","pages":"Pages 45-63"},"PeriodicalIF":1.0,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143683390","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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