Linear Algebra and its Applications最新文献

{"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}

{"title":"Quasi-isometric liftings for operators similar to contractions","authors":"Laurian Suciu,&nbsp;Andra-Maria Stoica","doi":"10.1016/j.laa.2025.01.006","DOIUrl":"10.1016/j.laa.2025.01.006","url":null,"abstract":"<div><div>A class of quasi-isometric liftings for the operators <em>T</em> similar to contractions in Hilbert spaces <span><math><mi>H</mi></math></span> is studied. These liftings are isometric operators on their ranges, and are naturally induced by <em>T</em> and an invertible intertwiner of <em>T</em> with a contraction. In the case when <em>T</em> is a quasicontraction, meaning that <em>T</em> is contractive on its range, we obtain a quasi-isometric lifting on a space <span><math><mi>K</mi><mo>⊃</mo><mi>H</mi></math></span>, which is isometric on <span><math><mi>K</mi><mo>⊖</mo><mi>H</mi></math></span>. Some liftings with closed ranges, or even similar to quasinormal partial isometries are mentioned. Additionally, we study the isomorphic minimal quasi-isometric liftings for <em>T</em>, as well as the uniqueness property of such liftings. Our results show similarities with those from the isometric dilation theory for contractions, although our context is more general than that of the latter.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"709 ","pages":"Pages 40-57"},"PeriodicalIF":1.0,"publicationDate":"2025-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143130287","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 triangulant","authors":"Tamás Bencze ,&nbsp;Péter E. Frenkel","doi":"10.1016/j.laa.2025.01.004","DOIUrl":"10.1016/j.laa.2025.01.004","url":null,"abstract":"<div><div>We introduce the triangulant of two matrices, and relate it to the existence of orthogonal eigenvectors. We also use it for a new characterization of mutually unbiased bases. Generalizing the notion, we introduce higher order triangulants of two matrices, and relate them to the existence of nontrivially intersecting invariant subspaces of complementary dimensions.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"709 ","pages":"Pages 92-110"},"PeriodicalIF":1.0,"publicationDate":"2025-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143130289","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":"Quadratic embedding constants of fan graphs and graph joins","authors":"Wojciech Młotkowski ,&nbsp;Nobuaki Obata","doi":"10.1016/j.laa.2025.01.001","DOIUrl":"10.1016/j.laa.2025.01.001","url":null,"abstract":"<div><div>We derive a general formula for the quadratic embedding constant of a graph join <span><math><msub><mrow><mover><mrow><mi>K</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mi>m</mi></mrow></msub><mo>+</mo><mi>G</mi></math></span>, where <span><math><msub><mrow><mover><mrow><mi>K</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mi>m</mi></mrow></msub></math></span> is the empty graph on <span><math><mi>m</mi><mo>≥</mo><mn>1</mn></math></span> vertices and <em>G</em> is an arbitrary graph. Applying our formula to a fan graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, where <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><msub><mrow><mover><mrow><mi>K</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mn>1</mn></mrow></msub></math></span> is the singleton graph and <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is the path on <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span> vertices, we show that <span><math><mrow><mi>QEC</mi></mrow><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>=</mo><mo>−</mo><msub><mrow><mover><mrow><mi>α</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>n</mi></mrow></msub><mo>−</mo><mn>2</mn></math></span>, where <span><math><msub><mrow><mover><mrow><mi>α</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>n</mi></mrow></msub></math></span> is the minimal zero of a new polynomial <span><math><msub><mrow><mi>Φ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> related to Chebyshev polynomials of the second kind. Moreover, for an even <em>n</em> we have <span><math><msub><mrow><mover><mrow><mi>α</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>min</mi><mo>⁡</mo><mrow><mi>ev</mi></mrow><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span>, where the right-hand side is the minimal eigenvalue of the adjacency matrix <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. For an odd <em>n</em> we show that <span><math><mi>min</mi><mo>⁡</mo><mrow><mi>ev</mi></mrow><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo><mo>≤</mo><msub><mrow><mover><mrow><mi>α</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>n</mi></mrow></msub><mo>&lt;</mo><mi>min</mi><mo>⁡</mo><mrow><mi>ev</mi></mrow><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"709 ","pages":"Pages 58-91"},"PeriodicalIF":1.0,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143130286","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}