{"title":"A new generalization of Fielder's lemma with applications","authors":"Komal Kumari, 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}
{"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}
{"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":"Asymptotic analysis of generalized orthogonal flows","authors":"Yueh-Cheng Kuo , Huey-Er Lin , 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}
{"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 some conjectures concerning the Böttcher-Wenzel inequality for weighted Frobenius norms","authors":"Wenbo Fang, 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}
{"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}
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