{"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}
Shaun Fallat , Himanshu Gupta , Charles R. Johnson
{"title":"Sufficient conditions for total positivity, compounds, and Dodgson condensation","authors":"Shaun Fallat , Himanshu Gupta , Charles R. Johnson","doi":"10.1016/j.laa.2025.01.016","DOIUrl":"10.1016/j.laa.2025.01.016","url":null,"abstract":"<div><div>A <em>n</em>-by-<em>n</em> matrix is called totally positive (<em>TP</em>) if all its minors are positive and <span><math><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> if all of its <em>k</em>-by-<em>k</em> submatrices are <em>TP</em>. For an arbitrary totally positive matrix or <span><math><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> matrix, we investigate if the <em>r</em>th compound (<span><math><mn>1</mn><mo><</mo><mi>r</mi><mo><</mo><mi>n</mi></math></span>) is in turn <em>TP</em> or <span><math><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, and demonstrate a strong negative resolution in general. Focus is then shifted to Dodgson's algorithm for calculating the determinant of a generic matrix, and we analyze whether the associated condensed matrices are possibly totally positive or <span><math><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>. We also show that all condensed matrices associated with a <em>TP</em> Hankel matrix are <em>TP</em>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"709 ","pages":"Pages 189-202"},"PeriodicalIF":1.0,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143130292","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":"Characterizations of homomorphisms among unital completely positive maps","authors":"Andre Kornell","doi":"10.1016/j.laa.2025.01.014","DOIUrl":"10.1016/j.laa.2025.01.014","url":null,"abstract":"<div><div>We prove that a unital completely positive map between finite-dimensional <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebras is a homomorphism if and only if it is completely entropy-nonincreasing, where the relevant notion of entropy is a variant of von Neumann entropy. This adjusted von Neumann entropy is the negative of the relative entropy with respect to the uniform state on the <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebra, up to an additive constant. As an intermediate step, we prove that a unital completely positive map between finite-dimensional <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebras is a homomorphism if and only if its adjusted Choi operator is a projection. Both equivalences generalize familiar facts about stochastic maps between finite sets.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"709 ","pages":"Pages 314-330"},"PeriodicalIF":1.0,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143129884","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 marginal growth rates of matrix products","authors":"Jonah Varney , Ian D. Morris","doi":"10.1016/j.laa.2025.01.013","DOIUrl":"10.1016/j.laa.2025.01.013","url":null,"abstract":"<div><div>In this article we consider the maximum possible growth rate of sequences of long products of <span><math><mi>d</mi><mo>×</mo><mi>d</mi></math></span> matrices all of which are drawn from some specified compact set which has been normalised so as to have joint spectral radius equal to 1. We define the <em>marginal instability rate sequence</em> associated to such a set to be the sequence of real numbers whose <span><math><msup><mrow><mi>n</mi></mrow><mrow><mi>t</mi><mi>h</mi></mrow></msup></math></span> entry is the norm of the largest product of length <em>n</em>, and study the general properties of sequences of this form. We describe how new marginal instability rate sequences can be constructed from old ones, extend an earlier example of Protasov and Jungers to obtain marginal instability rate sequences whose limit superior rate of growth matches various non-integer powers of <em>n</em>, and investigate the relationship between marginal instability rate sequences arising from finite sets of matrices and those arising from sets of matrices with cardinality 2. We also give the first example of a finite set whose marginal instability rate sequence is asymptotically similar to a polynomial with non-integer exponent. Previous examples had this property only along a subsequence.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"709 ","pages":"Pages 132-163"},"PeriodicalIF":1.0,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143129843","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":"The height of an infinite parallelotope is infinite","authors":"Alexandr V. Kosyak","doi":"10.1016/j.laa.2025.01.011","DOIUrl":"10.1016/j.laa.2025.01.011","url":null,"abstract":"<div><div>We show that if no non-trivial linear combinations of independent vectors <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span> belongs to <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, then all the heights of an infinite parallelotope constructed on vectors <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> are infinite. This result is essential in the proof of the irreducibility of unitary representations of some infinite-dimensional groups.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"709 ","pages":"Pages 18-39"},"PeriodicalIF":1.0,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143130258","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":"When is every linear transformation a sum of a q-potent one and a locally nilpotent one?","authors":"A.N. Abyzov, D.T. Tapkin","doi":"10.1016/j.laa.2025.01.012","DOIUrl":"10.1016/j.laa.2025.01.012","url":null,"abstract":"<div><div>We prove that for each vector space <em>V</em> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>, every linear transformation of <em>V</em> is a sum of a <em>q</em>-potent linear transformation and a locally nilpotent linear transformation.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"709 ","pages":"Pages 124-131"},"PeriodicalIF":1.0,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143130288","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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