Linear Algebra and its Applications最新文献

{"title":"Indefinite determinantal representations versus nonsingularities on the noncommutative d-torus","authors":"Gilbert J. Groenewald ,&nbsp;Sanne ter Horst ,&nbsp;Hugo J. Woerdeman","doi":"10.1016/j.laa.2025.04.024","DOIUrl":"10.1016/j.laa.2025.04.024","url":null,"abstract":"<div><div>We show that for a multivariable polynomial <span><math><mi>p</mi><mo>(</mo><mi>z</mi><mo>)</mo><mo>=</mo><mi>p</mi><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>d</mi></mrow></msub><mo>)</mo></math></span> with a determinantal representation<span><span><span><math><mi>p</mi><mo>(</mo><mi>z</mi><mo>)</mo><mo>=</mo><mi>p</mi><mo>(</mo><mn>0</mn><mo>)</mo><mi>det</mi><mo>⁡</mo><mo>(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>−</mo><mi>K</mi><mo>(</mo><msubsup><mrow><mo>⊕</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>d</mi></mrow></msubsup><msub><mrow><mi>z</mi></mrow><mrow><mi>j</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msub><mo>)</mo><mo>)</mo></math></span></span></span> the matrix <em>K</em> is structurally similar to a strictly <em>J</em>-contractive matrix for some diagonal signature matrix <em>J</em> if and only if the extension of <span><math><mi>p</mi><mo>(</mo><mi>z</mi><mo>)</mo></math></span> to a polynomial in <em>d</em>-tuples of matrices of arbitrary size given by<span><span><span><math><mi>p</mi><mo>(</mo><msub><mrow><mi>U</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo><mo>=</mo><mi>p</mi><mo>(</mo><mn>0</mn><mo>)</mo><mi>det</mi><mo>⁡</mo><mo>(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>⊗</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>−</mo><mo>(</mo><mi>K</mi><mo>⊗</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo><mo>(</mo><msubsup><mrow><mo>⊕</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>d</mi></mrow></msubsup><msub><mrow><mi>I</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msub><mo>⊗</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo><mo>)</mo><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mi>U</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>m</mi><mo>×</mo><mi>m</mi></mrow></msup></math></span>, <span><math><mi>m</mi><mo>∈</mo><mi>N</mi></math></span>, does not have roots on the noncommutative <em>d</em>-torus consisting of <em>d</em>-tuples <span><math><mo>(</mo><msub><mrow><mi>U</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo></math></span> of unitary matrices of arbitrary size.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"720 ","pages":"Pages 245-255"},"PeriodicalIF":1.0,"publicationDate":"2025-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143937694","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 stability of pure point spectrum of discrete operators with long-range hopping","authors":"Wenwen Jian ,&nbsp;Li Wen","doi":"10.1016/j.laa.2025.04.021","DOIUrl":"10.1016/j.laa.2025.04.021","url":null,"abstract":"<div><div>Let <em>T</em> be a multiplication operator on <span><math><msup><mrow><mi>ℓ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> with pure-point and simple spectrum. In this paper, we study perturbation of <em>T</em> by some off-diagonal Toeplitz operator <em>V</em> with its matrix elements <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span> satisfying <span><math><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>|</mo><mo>≤</mo><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mi>r</mi><msup><mrow><mi>log</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>⁡</mo><mo>(</mo><mn>1</mn><mo>+</mo><mo>|</mo><mi>m</mi><mo>−</mo><mi>n</mi><mo>|</mo><mo>)</mo></mrow></msup></math></span> <span><math><mo>(</mo><mi>t</mi><mo>&gt;</mo><mn>1</mn><mo>,</mo><mi>r</mi><mo>&gt;</mo><mn>0</mn><mo>,</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>∈</mo><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span>. Under some explicit conditions on the eigenvalues of <em>T</em> and for small enough <em>ε</em>, we prove via a KAM diagonalization method that the operator <span><math><mi>T</mi><mo>+</mo><mi>ε</mi><mi>V</mi></math></span> has pure-point spectrum with eigenfunctions <span><math><msub><mrow><mo>{</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo><mo>}</mo></mrow><mrow><mi>n</mi><mo>∈</mo><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></mrow></msub></math></span> obeying <span><math><mo>|</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo><mo>|</mo><mo>≤</mo><mn>2</mn><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mfrac><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></mfrac><msup><mrow><mi>log</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>⁡</mo><mo>(</mo><mn>1</mn><mo>+</mo><mo>|</mo><mi>k</mi><mo>−</mo><mi>n</mi><mo>|</mo><mo>)</mo></mrow></msup></math></span> for all <span><math><mi>n</mi><mo>,</mo><mi>k</mi><mo>∈</mo><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"720 ","pages":"Pages 152-173"},"PeriodicalIF":1.0,"publicationDate":"2025-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143937716","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":"Variants of expansivity in linear dynamics","authors":"Priyabrata Bag ,&nbsp;Pramod Kumar Das ,&nbsp;Hiroyuki Osaka ,&nbsp;Shubhankar Podder","doi":"10.1016/j.laa.2025.04.025","DOIUrl":"10.1016/j.laa.2025.04.025","url":null,"abstract":"<div><div>We study various topological dynamical notions, including positive expansivity, expansivity, and measure-expansivity for linear operators. We provide examples to show that these notions are distinct from each other, even for linear operators. We provide sufficient conditions, in terms of the restricted operators on suitable subspaces of the phase space, for the linear operators to be positive expansive, expansive, and measure-expansive. For some special operators, we provide necessary and sufficient conditions for measure-expansivity.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"720 ","pages":"Pages 256-271"},"PeriodicalIF":1.0,"publicationDate":"2025-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143937695","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":"An isospectral transformation between Hessenberg–bidiagonal matrix pencils and Hessenberg matrices without using subtraction","authors":"Katsuki Kobayashi ,&nbsp;Kazuki Maeda ,&nbsp;Satoshi Tsujimoto","doi":"10.1016/j.laa.2025.04.022","DOIUrl":"10.1016/j.laa.2025.04.022","url":null,"abstract":"<div><div>We introduce an eigenvalue-preserving transformation algorithm from the generalized eigenvalue problem by matrix pencil of the upper and the lower bidiagonal matrices into a standard eigenvalue problem while preserving sparsity, using the theory of orthogonal polynomials. The procedure is formulated without subtraction, which causes numerical instability. Furthermore, the algorithm is discussed for the extended case where the upper bidiagonal matrix is of Hessenberg type.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"720 ","pages":"Pages 272-302"},"PeriodicalIF":1.0,"publicationDate":"2025-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143937696","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 linear targeting problem","authors":"Kyle Bierly ,&nbsp;Stephan Ramon Garcia ,&nbsp;Roger A. Horn","doi":"10.1016/j.laa.2025.04.017","DOIUrl":"10.1016/j.laa.2025.04.017","url":null,"abstract":"<div><div>For given real or complex <span><math><mi>m</mi><mo>×</mo><mi>n</mi></math></span> data matrices <em>X</em>, <em>Y</em>, we investigate when there is a matrix <em>A</em> such that <span><math><mi>A</mi><mi>X</mi><mo>=</mo><mi>Y</mi></math></span>, and <em>A</em> is invertible, Hermitian, positive (semi)definite, unitary, an orthogonal projection, a reflection, complex symmetric, or normal.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"720 ","pages":"Pages 91-108"},"PeriodicalIF":1.0,"publicationDate":"2025-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143908203","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 tropical variety of symmetric rank 2 matrices","authors":"May Cai ,&nbsp;Kisun Lee ,&nbsp;Josephine Yu","doi":"10.1016/j.laa.2025.04.011","DOIUrl":"10.1016/j.laa.2025.04.011","url":null,"abstract":"<div><div>We study the tropicalization of the variety of symmetric rank two matrices. Analogously to the result of Markwig and Yu for general tropical rank two matrices, we show that it has a simplicial complex structure as the space of symmetric bicolored trees and that this simplicial complex is shellable. We also discuss some matroid structures arising from this space and present generating functions for the number of symmetric bicolored trees.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"720 ","pages":"Pages 50-71"},"PeriodicalIF":1.0,"publicationDate":"2025-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143888268","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":"Extensions on spectral extrema of θ1,2,5-free graphs with given size","authors":"Chang Liu,&nbsp;Jianping Li","doi":"10.1016/j.laa.2025.04.010","DOIUrl":"10.1016/j.laa.2025.04.010","url":null,"abstract":"<div><div>The Brualdi-Hoffman-Turán type problem is an important category of spectral Turán problems, focusing on determining the maximum spectral radius of an <span><math><mi>F</mi></math></span>-free graph with <em>m</em> edges. This topic has received considerable attention in recent years. Let <span><math><mi>G</mi><mo>(</mo><mi>m</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> denote the set of <span><math><mi>F</mi></math></span>-free graphs with <em>m</em> edges and no isolated vertices. Define <span><math><msub><mrow><mi>θ</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi>r</mi></mrow></msub></math></span> as the theta graph, consisting of three internally disjoint paths of lengths <em>p</em>, <em>q</em>, and <em>r</em>, sharing the same pair of endpoints. Recently, Lu et al. (2024) <span><span>[16]</span></span> determined that the unique extremal graph with the maximum spectral radius in <span><math><mi>G</mi><mo>(</mo><mi>m</mi><mo>,</mo><msub><mrow><mi>θ</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>5</mn></mrow></msub><mo>)</mo></math></span> is <span><math><msub><mrow><mi>S</mi></mrow><mrow><mfrac><mrow><mi>m</mi><mo>+</mo><mn>6</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>,</mo><mn>3</mn></mrow></msub></math></span> for <span><math><mi>m</mi><mo>≥</mo><mn>38</mn></math></span>. However, the extremal graph is well-defined only when <span><math><mi>m</mi><mo>≡</mo><mn>0</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>3</mn><mo>)</mo></math></span>. In this paper, we employ a new technique to characterize the graphs with the maximum spectral radius among all <span><math><msub><mrow><mi>θ</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>5</mn></mrow></msub></math></span>-free graphs for <span><math><mi>m</mi><mo>≥</mo><mn>39</mn></math></span>, where <span><math><mi>m</mi><mo>≡</mo><mn>1</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>3</mn><mo>)</mo></math></span> or <span><math><mi>m</mi><mo>≡</mo><mn>2</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>3</mn><mo>)</mo></math></span>. Finally, we propose two conjectures that generalize Zhai et al. (2021) <span><span>[31]</span></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"719 ","pages":"Pages 136-157"},"PeriodicalIF":1.0,"publicationDate":"2025-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143886056","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":"Complete characterization of symmetric Kubo-Ando operator means satisfying Molnár's weak associativity","authors":"Yury Grabovsky ,&nbsp;Graeme W. Milton ,&nbsp;Aaron Welters","doi":"10.1016/j.laa.2025.04.015","DOIUrl":"10.1016/j.laa.2025.04.015","url":null,"abstract":"<div><div>We provide a complete characterization of a subclass of weakly associative means of positive operators in the class of symmetric Kubo-Ando means. This class, which includes the geometric mean, was first introduced and studied in Molnár (2019) <span><span>[24]</span></span>, where he gives a characterization of this subclass (which we call the Molnár class of means) in terms of the properties of their representing operator monotone functions. Molnár's paper leaves open the problem of determining if the geometric mean is the only such mean in that subclass. Here we give a negative answer to this question by constructing an order-preserving bijection between this class and a class of real measurable odd periodic functions bounded in absolute value by 1/2. Each member of the latter class defines a Molnár mean by an explicit exponential-integral representation. From this we are able to understand the order structure of the Molnár class and construct several infinite families of explicit examples of Molnár means that are not the geometric mean. Our analysis also shows how to modify Molnár's original characterization so that the geometric mean is the only one satisfying the requisite set of properties.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"719 ","pages":"Pages 158-182"},"PeriodicalIF":1.0,"publicationDate":"2025-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143894761","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":"Partial classification of spectrum maximizing products for pairs of 2 × 2 matrices","authors":"Piotr Laskawiec","doi":"10.1016/j.laa.2025.04.014","DOIUrl":"10.1016/j.laa.2025.04.014","url":null,"abstract":"<div><div>Experiments suggest that typical finite sets of square matrices admit spectrum maximizing products (SMPs): that is, products that attain the joint spectral radius (JSR). Furthermore, those SMPs are often combinatorially “simple.” In this paper, we consider pairs of real <span><math><mn>2</mn><mo>×</mo><mn>2</mn></math></span> matrices. We identify regions in the space of such pairs where SMPs are guaranteed to exist and to have a simple structure. We also identify another region where SMPs may fail to exist (in fact, this region includes all known counterexamples to the finiteness conjecture), but nevertheless a Sturmian maximizing measure exists. Though our results apply to a large chunk of the space of pairs of <span><math><mn>2</mn><mo>×</mo><mn>2</mn></math></span> matrices, including for instance all pairs of non-negative matrices, they leave out certain “wild” regions where more complicated behavior is possible.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"719 ","pages":"Pages 103-135"},"PeriodicalIF":1.0,"publicationDate":"2025-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143886055","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":"Maximization of the first Laplace eigenvalue of a finite graph","authors":"Takumi Gomyou ,&nbsp;Shin Nayatani","doi":"10.1016/j.laa.2025.04.013","DOIUrl":"10.1016/j.laa.2025.04.013","url":null,"abstract":"<div><div>Given a length function on the edge set of a finite graph, we define a vertex-weight and an edge-weight in terms of it and consider the corresponding graph Laplacian. In this paper, we consider the problem of maximizing the first nonzero eigenvalue of this Laplacian over all edge-length functions subject to a certain normalization. For an extremal solution of this problem, we prove that there exists a map from the vertex set to a Euclidean space consisting of first eigenfunctions of the corresponding Laplacian so that the length function can be explicitly expressed in terms of the map and the Euclidean distance. This is a graph-analogue of Nadirashvili's result related to first-eigenvalue maximization problem on a smooth surface. We discuss simple examples and also prove a similar result for a maximizing solution of the Göring-Helmberg-Wappler problem.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"720 ","pages":"Pages 72-90"},"PeriodicalIF":1.0,"publicationDate":"2025-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143912831","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}