Linear Algebra and its Applications最新文献

{"title":"Sufficient conditions for total positivity, compounds, and Dodgson condensation","authors":"Shaun Fallat ,&nbsp;Himanshu Gupta ,&nbsp;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>&lt;</mo><mi>r</mi><mo>&lt;</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 ,&nbsp;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,&nbsp;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}

{"title":"Corrigendum to “A matricial view of the Collatz conjecture” [Linear Algebra Appl. 695 (2024) 163–167]","authors":"Pietro Paparella","doi":"10.1016/j.laa.2024.12.019","DOIUrl":"10.1016/j.laa.2024.12.019","url":null,"abstract":"<div><div>There is a mistake in the proof of <span><span>Theorem 3</span></span> of “A matricial view of the Collatz conjecture” <span><span>[1]</span></span> that can not be rectified. As such, a revised statement and proof of <span><span>Theorem 3</span></span> is presented.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 608-609"},"PeriodicalIF":1.0,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164636","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":"Darboux transformations and the algebra D(W)","authors":"Ignacio Bono Parisi,&nbsp;Ines Pacharoni","doi":"10.1016/j.laa.2025.01.002","DOIUrl":"10.1016/j.laa.2025.01.002","url":null,"abstract":"<div><div>The problem of finding weight matrices <span><math><mi>W</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> of size <span><math><mi>N</mi><mo>×</mo><mi>N</mi></math></span> such that the associated sequence of matrix-valued orthogonal polynomials are eigenfunctions of a second-order matrix differential operator is known as the Matrix Bochner Problem, and it is closely related to Darboux transformations of some differential operators.</div><div>This paper aims to study Darboux transformations between weight matrices and to establish a direct connection with the structure of the algebra <span><math><mi>D</mi><mo>(</mo><mi>W</mi><mo>)</mo></math></span> of all differential operators that have a sequence of matrix-valued orthogonal polynomials with respect to <em>W</em> as eigenfunctions.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"709 ","pages":"Pages 203-232"},"PeriodicalIF":1.0,"publicationDate":"2025-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143130293","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":"Mixed tensor invariants of Lie color algebra","authors":"Santosha Pattanayak,&nbsp;Preena Samuel","doi":"10.1016/j.laa.2025.01.003","DOIUrl":"10.1016/j.laa.2025.01.003","url":null,"abstract":"<div><div>In this paper, we consider the mixed tensor space of a <em>G</em>-graded vector space, where <em>G</em> is a finite abelian group. We obtain a spanning set of invariants of the associated symmetric algebra under the action of a color analogue of the general linear group which we refer to as the general linear color group. As a consequence, we obtain a generating set for the polynomial invariants, under the simultaneous action of the general linear color group, on color analogues of several copies of matrices. We show that in this special case, this is the set of trace monomials, which coincides with the set of generators given by Berele in <span><span>[2]</span></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"709 ","pages":"Pages 1-17"},"PeriodicalIF":1.0,"publicationDate":"2025-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143130259","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 methods for matrix product factorization","authors":"Saieed Akbari ,&nbsp;Yi-Zheng Fan ,&nbsp;Fu-Tao Hu ,&nbsp;Babak Miraftab ,&nbsp;Yi Wang","doi":"10.1016/j.laa.2025.01.005","DOIUrl":"10.1016/j.laa.2025.01.005","url":null,"abstract":"<div><div>A graph <em>G</em> is factored into graphs <em>H</em> and <em>K</em> via a matrix product if there exist adjacency matrices <em>A</em>, <em>B</em>, and <em>C</em> of <em>G</em>, <em>H</em>, and <em>K</em>, respectively, such that <span><math><mi>A</mi><mo>=</mo><mi>B</mi><mi>C</mi></math></span>. In this paper, we study the spectral aspects of the matrix product of graphs, including regularity, bipartiteness, and connectivity. We show that if a graph <em>G</em> is factored into a connected graph <em>H</em> and a graph <em>K</em> with no isolated vertices, then certain properties hold. If <em>H</em> is non-bipartite, then <em>G</em> is connected. If <em>H</em> is bipartite and <em>G</em> is not connected, then <em>K</em> is a regular bipartite graph, and consequently, <em>n</em> is even. Furthermore, we show that trees are not factorizable, which answers a question posed by Maghsoudi et al.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"709 ","pages":"Pages 111-123"},"PeriodicalIF":1.0,"publicationDate":"2025-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143130290","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":"Refined inertias of nonnegative patterns with positive off-diagonal entries","authors":"Adam H. Berliner ,&nbsp;Minerva Catral ,&nbsp;D.D. Olesky ,&nbsp;P. van den Driessche","doi":"10.1016/j.laa.2025.01.008","DOIUrl":"10.1016/j.laa.2025.01.008","url":null,"abstract":"<div><div>For a positive <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> pattern <span><math><mi>A</mi></math></span>, it is known that the refined inertia of <span><math><mi>A</mi></math></span>, <span><math><mi>ri</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, is the set of all nonnegative integral 4-tuples <span><math><mo>(</mo><msub><mrow><mi>n</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mo>−</mo></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>,</mo><mn>2</mn><msub><mrow><mi>n</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math></span> with <span><math><msub><mrow><mi>n</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mo>−</mo></mrow></msub><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>+</mo><mn>2</mn><msub><mrow><mi>n</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>=</mo><mi>n</mi></math></span> and <span><math><msub><mrow><mi>n</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>≥</mo><mn>1</mn></math></span>; whereas if <span><math><mi>A</mi></math></span> has all off-diagonal entries positive but all diagonal entries 0, then <span><math><mi>ri</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> has the additional restriction that <span><math><msub><mrow><mi>n</mi></mrow><mrow><mo>−</mo></mrow></msub><mo>≥</mo><mn>2</mn></math></span>. We focus on the intermediate nonnegative patterns, that is those patterns with all off-diagonal entries positive, <span><math><mi>k</mi><mo>∈</mo><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>}</mo></math></span> diagonal entries positive and the remaining <span><math><mi>n</mi><mo>−</mo><mi>k</mi></math></span> diagonal entries 0. We show that for <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span>, there is no restriction on <span><math><msub><mrow><mi>n</mi></mrow><mrow><mo>−</mo></mrow></msub></math></span> for the refined inertia set, but <span><math><msub><mrow><mi>n</mi></mrow><mrow><mo>−</mo></mrow></msub><mo>≥</mo><mn>1</mn></math></span> for <span><math><mi>k</mi><mo>=</mo><mn>1</mn></math></span>. We do this by constructing nonnegative matrix realizations for the patterns with <span><math><mi>k</mi><mo>=</mo><mn>1</mn></math></span> and 2 using the centralizer method, matrix bordering and superpattern results.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"709 ","pages":"Pages 271-283"},"PeriodicalIF":1.0,"publicationDate":"2025-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143130255","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}