{"title":"Multiple typical ranks in matrix completion","authors":"Mareike Dressler , Robert Krone","doi":"10.1016/j.laa.2025.01.026","DOIUrl":"10.1016/j.laa.2025.01.026","url":null,"abstract":"<div><div>Low-rank matrix completion addresses the problem of completing a matrix from a certain set of generic specified entries. Over the complex numbers a matrix with a given entry pattern can be uniquely completed to a specific rank, called the generic completion rank. Completions over the reals may generically have multiple completion ranks, called typical ranks. We demonstrate techniques for proving that many sets of specified entries have only one typical rank, and show other families with two typical ranks, specifically focusing on entry sets represented by circulant graphs. This generalizes the results of Bernstein, Blekherman, and Sinn. In particular, we provide a complete characterization of the set of unspecified entries of an <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrix such that <span><math><mi>n</mi><mo>−</mo><mn>1</mn></math></span> is a typical rank and fully determine the typical ranks of an <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrix with unspecified diagonal for <span><math><mi>n</mi><mo><</mo><mn>9</mn></math></span>. Moreover, we study the asymptotic behavior of typical ranks and present results regarding unique matrix completions.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 165-182"},"PeriodicalIF":1.0,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143277786","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}
Bojan Kuzma , Chi-Kwong Li , Edward Poon , Sushil Singla
{"title":"Linear preservers of parallel matrix pairs with respect to the k-numerical radius","authors":"Bojan Kuzma , Chi-Kwong Li , Edward Poon , Sushil Singla","doi":"10.1016/j.laa.2025.01.019","DOIUrl":"10.1016/j.laa.2025.01.019","url":null,"abstract":"<div><div>Let <span><math><mn>1</mn><mo>≤</mo><mi>k</mi><mo><</mo><mi>n</mi></math></span> be integers. Two <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices <em>A</em> and <em>B</em> form a parallel pair with respect to the <em>k</em>-numerical radius <span><math><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> if <span><math><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>+</mo><mi>μ</mi><mi>B</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>B</mi><mo>)</mo></math></span> for some scalar <em>μ</em> with <span><math><mo>|</mo><mi>μ</mi><mo>|</mo><mo>=</mo><mn>1</mn></math></span>; they form a TEA (triangle equality attaining) pair if the preceding equation holds for <span><math><mi>μ</mi><mo>=</mo><mn>1</mn></math></span>. We classify linear bijections on <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and on <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> which preserve parallel pairs or TEA pairs. Such preservers are scalar multiples of <span><math><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>-isometries, except for some exceptional maps on <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> when <span><math><mi>n</mi><mo>=</mo><mn>2</mn><mi>k</mi></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"709 ","pages":"Pages 342-363"},"PeriodicalIF":1.0,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143129883","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 error momentum Bregman-Kaczmarz","authors":"Dirk A. Lorenz, Maximilian Winkler","doi":"10.1016/j.laa.2025.01.024","DOIUrl":"10.1016/j.laa.2025.01.024","url":null,"abstract":"<div><div>The Bregman-Kaczmarz method is an iterative method which can solve strongly convex problems with linear constraints and uses only one or a selected number of rows of the system matrix in each iteration, thereby making it amenable for large-scale systems. To speed up convergence, we investigate acceleration by heavy ball momentum in the so-called dual update. Heavy ball acceleration of the Kaczmarz method with constant parameters has turned out to be difficult to analyze, in particular no accelerated convergence for the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-error of the iterates has been proven to the best of our knowledge. Here we propose a way to adaptively choose the momentum parameter by a minimal-error principle similar to a recently proposed method for the standard randomized Kaczmarz method. The momentum parameter can be chosen to exactly minimize the error in the next iterate or to minimize a relaxed version of the minimal error principle. The former choice leads to a theoretically optimal step while the latter is cheaper to compute. We prove improved convergence results compared to the non-accelerated method. Numerical experiments show that the proposed methods can accelerate convergence in practice, also for matrices which arise from applications such as computational tomography.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"709 ","pages":"Pages 416-448"},"PeriodicalIF":1.0,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143129886","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":"Maximum Aα-spectral radius of {C(3,3),C(4,3)}-free graphs","authors":"S. Pirzada, Amir Rehman","doi":"10.1016/j.laa.2025.01.023","DOIUrl":"10.1016/j.laa.2025.01.023","url":null,"abstract":"<div><div>For a simple graph <em>G</em> and for any <span><math><mi>α</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span>, Nikiforov defined the generalized adjacency matrix as <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>α</mi><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mo>(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo>)</mo><mi>A</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, where <span><math><mi>A</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and <span><math><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> are the adjacency and degree diagonal matrices of <em>G</em>, respectively. The largest eigenvalue of <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is called the generalized adjacency spectral radius (or <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>-spectral radius) of <em>G</em>. Let <span><math><mi>C</mi><mo>(</mo><mi>l</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> denote the graph obtained from <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>l</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> by superimposing an edge of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>l</mi></mrow></msub></math></span> with an edge of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span>. If a graph is free of both <span><math><mi>C</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>3</mn><mo>)</mo></math></span> and <span><math><mi>C</mi><mo>(</mo><mn>4</mn><mo>,</mo><mn>3</mn><mo>)</mo></math></span>, we call it a <span><math><mo>{</mo><mi>C</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>,</mo><mi>C</mi><mo>(</mo><mn>4</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>}</mo></math></span>-free graph. In this paper, we give a sharp upper bound on the <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>-spectral radius of <span><math><mo>{</mo><mi>C</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>,</mo><mi>C</mi><mo>(</mo><mn>4</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>}</mo></math></span>-free graphs for <span><math><mi>α</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></math></span>. We show that the extremal graph attaining the bound is the 2-partite Turán graph.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"709 ","pages":"Pages 385-396"},"PeriodicalIF":1.0,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143129893","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":"Some families of digraphs determined by the complementarity spectrum","authors":"Diego Bravo , Florencia Cubría , Marcelo Fiori , Gustavo Rama","doi":"10.1016/j.laa.2025.01.022","DOIUrl":"10.1016/j.laa.2025.01.022","url":null,"abstract":"<div><div>In this paper we study seven families of digraphs, and we determine whether digraphs in these families can be determined by their spectral radius. These seven families have been characterized as the only families of digraphs with exactly three complementarity eigenvalues <span><span>[1]</span></span>, and therefore our results have consequences in this context, showing which families can be determined by the complementarity spectrum. As a particular case, we prove that the <em>θ</em>-digraphs can be characterized by the spectral radius, extending some recent results on this family <span><span>[2]</span></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"709 ","pages":"Pages 364-384"},"PeriodicalIF":1.0,"publicationDate":"2025-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143129895","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":"Additive maps preserving rank-bounded sets of matrices","authors":"E. Akhmedova , A. Guterman , I. Spiridonov","doi":"10.1016/j.laa.2025.01.018","DOIUrl":"10.1016/j.laa.2025.01.018","url":null,"abstract":"<div><div>Let <span><math><mn>2</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></math></span> be integers and <span><math><msub><mrow><mi>Mat</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math></span> be the linear space of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices over a field <span><math><mi>F</mi></math></span> of characteristic different from 2. Denote by <span><math><msup><mrow><mi>Γ</mi></mrow><mrow><mo>≥</mo><mi>k</mi></mrow></msup></math></span> the set of matrices in <span><math><msub><mrow><mi>Mat</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math></span> of rank greater than or equal to <em>k</em>. The main goal of the present paper is to obtain a characterization of additive maps <span><math><mi>f</mi><mo>:</mo><msub><mrow><mi>Mat</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo><mo>→</mo><msub><mrow><mi>Mat</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math></span> satisfying <span><math><mi>f</mi><mo>(</mo><msup><mrow><mi>Γ</mi></mrow><mrow><mo>≥</mo><mi>k</mi></mrow></msup><mo>)</mo><mo>=</mo><msup><mrow><mi>Γ</mi></mrow><mrow><mo>≥</mo><mi>k</mi></mrow></msup></math></span> with either <span><math><mi>n</mi><mo><</mo><mn>2</mn><mi>k</mi><mo>−</mo><mn>2</mn></math></span> or <span><math><mi>F</mi></math></span> has characteristic <span><math><mrow><mi>char</mi><mspace></mspace></mrow><mo>(</mo><mi>F</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span> or <span><math><mrow><mi>char</mi><mspace></mspace></mrow><mo>(</mo><mi>F</mi><mo>)</mo><mo>≥</mo><mi>k</mi></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"709 ","pages":"Pages 331-341"},"PeriodicalIF":1.0,"publicationDate":"2025-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143129885","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 cone-preserving solution to a nonsymmetric Riccati equation","authors":"Emil Vladu, Anders Rantzer","doi":"10.1016/j.laa.2025.01.020","DOIUrl":"10.1016/j.laa.2025.01.020","url":null,"abstract":"<div><div>In this paper, we provide the following simple equivalent condition for a nonsymmetric Algebraic Riccati Equation to admit a stabilizing cone-preserving solution: an associated coefficient matrix must be stable. The result holds under the assumption that said matrix be cross-positive on a proper cone, and it both extends and completes a corresponding sufficient condition for nonnegative matrices in the literature. Further, key to showing the above is the following result which we also provide: in order for a monotonically increasing sequence of cone-preserving matrices to converge, it is sufficient to be bounded above in a single vectorial direction.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"709 ","pages":"Pages 449-459"},"PeriodicalIF":1.0,"publicationDate":"2025-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143129894","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 symmetric hollow integer matrices with eigenvalues bounded from below","authors":"Zilin Jiang (姜子麟)","doi":"10.1016/j.laa.2025.01.021","DOIUrl":"10.1016/j.laa.2025.01.021","url":null,"abstract":"<div><div>A hollow matrix is a square matrix whose diagonal entries are all equal to zero. Define <span><math><msup><mrow><mi>λ</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>=</mo><msup><mrow><mi>ρ</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>ρ</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>≈</mo><mn>2.01980</mn></math></span>, where <em>ρ</em> is the unique real root of <span><math><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>=</mo><mi>x</mi><mo>+</mo><mn>1</mn></math></span>. We show that for every <span><math><mi>λ</mi><mo><</mo><msup><mrow><mi>λ</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, there exists <span><math><mi>n</mi><mo>∈</mo><mi>N</mi></math></span> such that if a symmetric hollow integer matrix has an eigenvalue less than −<em>λ</em>, then one of its principal submatrices of order at most <em>n</em> does as well. However, the same conclusion does not hold for any <span><math><mi>λ</mi><mo>≥</mo><msup><mrow><mi>λ</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"709 ","pages":"Pages 233-240"},"PeriodicalIF":1.0,"publicationDate":"2025-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143130257","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 Nevanlinna formula for matrix Nevanlinna-Pick interpolation","authors":"Yury Dyukarev","doi":"10.1016/j.laa.2025.01.007","DOIUrl":"10.1016/j.laa.2025.01.007","url":null,"abstract":"<div><div>In this paper, we study the matrix Nevanlinna-Pick interpolation problem in the completely indeterminate case. We obtain an explicit formula for the resolvent matrix in terms of rational matrix functions of the first and second kind. Additionally, we describe the set of all solutions to the matrix Nevanlinna-Pick interpolation problem using linear fractional transformations applied to Nevanlinna pairs. This result can be viewed as an analogue of the Nevanlinna formula for the matrix Hamburger moment problem.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"709 ","pages":"Pages 241-270"},"PeriodicalIF":1.0,"publicationDate":"2025-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143129844","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":"Pareto singular values of Boolean matrices and analysis of bipartite graphs","authors":"Alberto Seeger , David Sossa","doi":"10.1016/j.laa.2025.01.015","DOIUrl":"10.1016/j.laa.2025.01.015","url":null,"abstract":"<div><div>The complementarity eigenvalues of a graph <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span>, which are defined as the Pareto eigenvalues of the adjacency matrix <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>G</mi></mrow></msub></math></span>, provide a rich information on structural properties of the graph. Complementarity eigenvalues are of special relevance for connected graphs. For instance, it has been conjectured that the complementarity eigenvalues of a connected graph determine the graph up to isomorphism. Analogously, the Pareto singular values of the biadjacency matrix <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>G</mi></mrow></msub></math></span> of a connected bipartite graph <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>U</mi><mo>,</mo><mi>W</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span> reflect various structural properties of the bipartite graph under consideration. The theory of Pareto singular values of general matrices was initiated in our paper entitled <em>Cone-constrained singular value problems</em> published in the Journal of Convex Analysis (30, 2023, pp. 1285-1306). In this work we explore various aspects of such a theory, paying special attention to Pareto singular values of Boolean matrices and their role in the analysis of bipartite graphs.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"709 ","pages":"Pages 164-188"},"PeriodicalIF":1.0,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143130291","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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