Linear Algebra and its Applications最新文献

{"title":"Threshold graphs, Kemeny's constant, and related random walk parameters","authors":"Jane Breen ,&nbsp;Sooyeong Kim ,&nbsp;Alexander Low Fung ,&nbsp;Amy Mann ,&nbsp;Andrei A. Parfeni ,&nbsp;Giovanni Tedesco","doi":"10.1016/j.laa.2024.12.022","DOIUrl":"10.1016/j.laa.2024.12.022","url":null,"abstract":"<div><div>Kemeny's constant measures how fast a random walker moves around in a graph. Expressions for Kemeny's constant can be quite involved, and for this reason, many lines of research focus on graphs with structure that makes them amenable to more in-depth study (for example, regular graphs, acyclic graphs, and 1-connected graphs). In this article, we study Kemeny's constant for random walks on threshold graphs, which are an interesting family of graphs with properties that make examining Kemeny's constant difficult; that is, they are usually not regular, not acyclic, and not 1-connected. This article is a showcase of various techniques for calculating Kemeny's constant and related random walk parameters for graphs. We establish explicit formulae for <span><math><mi>K</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> in terms of the construction code of a threshold graph, and completely determine the ordering of the accessibility indices of vertices in threshold graphs.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"709 ","pages":"Pages 284-313"},"PeriodicalIF":1.0,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143130256","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":"Symmetric doubly stochastic inverse eigenvalue problem for odd sizes","authors":"Mohadese Raeisi Sarkhoni ,&nbsp;Hossein Momenaee Kermani ,&nbsp;Azim Rivaz","doi":"10.1016/j.laa.2024.12.020","DOIUrl":"10.1016/j.laa.2024.12.020","url":null,"abstract":"<div><div>The symmetric doubly stochastic inverse eigenvalue problem seeks to determine the necessary and sufficient conditions for a real list of eigenvalues to be realized by a symmetric doubly stochastic matrix. Nader et al. (2019) <span><span>[15]</span></span>, established that for odd integers <em>n</em> a list of the form <span><math><mi>σ</mi><mo>=</mo><mo>(</mo><mn>1</mn><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><msub><mrow></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msub><mo>,</mo><mo>−</mo><mn>1</mn><mo>)</mo></math></span> with <span><math><mo>|</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo><mo>&lt;</mo><mn>1</mn></math></span> for <span><math><mi>i</mi><mo>=</mo><mn>2</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn></math></span> cannot be the spectrum of any <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> doubly stochastic matrix. This implies that the list <span><math><mi>σ</mi><mo>=</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mn>0</mn><mo>,</mo><mo>−</mo><mn>1</mn><mo>)</mo></math></span> is also unrealizable.</div><div>This paper extends these findings by proving that for odd <em>n</em> and <span><math><msub><mrow><mi>λ</mi></mrow><mrow><msub><mrow></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo>∈</mo><mo>[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mo>−</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mo>)</mo></math></span>, the list <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mn>0</mn><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><msub><mrow></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo>)</mo></math></span> cannot be the spectrum of a symmetric doubly stochastic matrix. We demonstrate that for odd <em>n</em> the list <span><math><mi>σ</mi><mo>=</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mn>0</mn><mo>,</mo><mo>−</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mo>)</mo></math></span>, is indeed realizable as the spectrum of a symmetric doubly stochastic matrix.</div><div>Furthermore, we utilize our methodology to derive new sufficient conditions for the existence of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> symmetric doubly stochastic matrices with a prescribed list of eigenvalues. This leads to a condition for the existence of symmetric doubly stochastic matrices with a normalized Suleimanova spectrum. The paper concludes with additional partial results and illustrative examples.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 594-607"},"PeriodicalIF":1.0,"publicationDate":"2024-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164638","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":"Linear bijective maps preserving invertibility on pairs of similar matrices","authors":"Constantin Costara","doi":"10.1016/j.laa.2024.12.021","DOIUrl":"10.1016/j.laa.2024.12.021","url":null,"abstract":"<div><div>Let <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span> be a natural number, and denote by <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> the space of all <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices over the complex field. In this paper, we characterize linear bijective maps <em>φ</em> on <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> having the property that if <span><math><mi>A</mi><mo>,</mo><mi>B</mi><mo>∈</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> are similar matrices and <span><math><mi>φ</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> is invertible, then <span><math><mi>φ</mi><mo>(</mo><mi>B</mi><mo>)</mo></math></span> is invertible as well.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 585-593"},"PeriodicalIF":1.0,"publicationDate":"2024-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143165108","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":"Asymptotic spectral properties and preconditioning of an approximated nonlocal Helmholtz equation with fractional Laplacian and variable coefficient wave number μ","authors":"Andrea Adriani ,&nbsp;Rosita L. Sormani ,&nbsp;Cristina Tablino-Possio ,&nbsp;Rolf Krause ,&nbsp;Stefano Serra-Capizzano","doi":"10.1016/j.laa.2024.12.015","DOIUrl":"10.1016/j.laa.2024.12.015","url":null,"abstract":"<div><div>The current study investigates the asymptotic spectral properties of a finite difference approximation of nonlocal Helmholtz equations with a fractional Laplacian and a variable coefficient wave number <em>μ</em>, as it occurs when considering a wave propagation in complex media, characterized by nonlocal interactions and spatially varying wave speeds. More specifically, by using tools from Toeplitz and generalized locally Toeplitz theory, the present research delves into the spectral analysis of nonpreconditioned and preconditioned matrix sequences, with the main novelty regarding a complete picture of the case where <span><math><mi>μ</mi><mo>=</mo><mi>μ</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> is nonconstant. We report numerical evidence supporting the theoretical findings. Finally, open problems and potential extensions in various directions are presented and briefly discussed.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 551-584"},"PeriodicalIF":1.0,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143165107","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":"Some stability results for spectral extremal problems of graphs with bounded matching number","authors":"Shixia Jiang ,&nbsp;Xiying Yuan ,&nbsp;Yanni Zhai","doi":"10.1016/j.laa.2024.12.018","DOIUrl":"10.1016/j.laa.2024.12.018","url":null,"abstract":"<div><div>For a set of graphs <span><math><mi>H</mi></math></span>, a graph is called <span><math><mi>H</mi></math></span>-free if it does not contain any member of <span><math><mi>H</mi></math></span> as a subgraph. The maximum value of spectral radius among all <span><math><mi>H</mi></math></span>-free graphs of order <em>n</em> is denoted by <span><math><mrow><mi>spex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span>, and the set of corresponding extremal graphs is denoted by <span><math><mrow><mi>SPEX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span>. In this paper, we give a stability result for graphs in <span><math><mrow><mi>SPEX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> when <span><math><mrow><mi>spex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo><mo>≥</mo><msqrt><mrow><mi>s</mi><mo>(</mo><mi>n</mi><mo>−</mo><mi>s</mi><mo>)</mo></mrow></msqrt></math></span> and <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo><mo>≤</mo><mi>s</mi><mi>n</mi></math></span>. As an application, we may give some characterizations for the graphs in <span><math><mrow><mi>SPEX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mo>{</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>H</mi><mo>}</mo><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> is a matching with <span><math><mi>s</mi><mo>+</mo><mn>1</mn></math></span> edges and <em>H</em> is any non-bipartite graph.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 513-524"},"PeriodicalIF":1.0,"publicationDate":"2024-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143165105","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":"Remarks on generating families of matrix algebras of small orders","authors":"Yaroslav Shitov","doi":"10.1016/j.laa.2024.12.013","DOIUrl":"10.1016/j.laa.2024.12.013","url":null,"abstract":"<div><div>Let <span><math><mi>n</mi><mo>⩽</mo><mn>7</mn></math></span>, and let <em>S</em> be a family of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices over a field <span><math><mi>F</mi></math></span>. I prove that the <span><math><mi>F</mi></math></span>-linear span of<span><span><span><math><msup><mrow><mo>(</mo><mi>S</mi><mo>∪</mo><mo>{</mo><mrow><mi>Id</mi></mrow><mo>}</mo><mo>)</mo></mrow><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msup></math></span></span></span> is the algebra generated by <em>S</em>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 458-462"},"PeriodicalIF":1.0,"publicationDate":"2024-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143165101","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":"Fractional perfect matching and distance spectral radius in graphs","authors":"Lei Zhang ,&nbsp;Yaoping Hou ,&nbsp;Haizhen Ren","doi":"10.1016/j.laa.2024.12.016","DOIUrl":"10.1016/j.laa.2024.12.016","url":null,"abstract":"<div><div>A fractional matching of a graph <em>G</em> is a function <em>f</em> giving each edge a number in <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span> so that <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>e</mi><mo>∈</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo></mrow></msub><mi>f</mi><mo>(</mo><mi>e</mi><mo>)</mo><mo>≤</mo><mn>1</mn></math></span> for each <span><math><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo></math></span> is the set of edges incident to <em>v</em>. In this paper, we give a distance spectral radius condition to guarantee the existence of a fractional perfect matching. This result generalize the result of Lin and Zhang (2021) <span><span>[22]</span></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 480-488"},"PeriodicalIF":1.0,"publicationDate":"2024-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143165103","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 necessary and sufficient conditions for Hadamard-Fischer-Koteljanskii type inequalities","authors":"Phillip Braun ,&nbsp;Hristo Sendov","doi":"10.1016/j.laa.2024.12.017","DOIUrl":"10.1016/j.laa.2024.12.017","url":null,"abstract":"<div><div>This work explores the ratios of products of determinants of principal submatrices of positive definite matrices. We investigate conditions under which these ratios are bounded, particularly revisiting the necessary/sufficient conditions proposed by Johnson and Barrett. This analysis extends to set-theoretic consequences and unboundedness of certain ratios. We also demonstrate how these conditions can be used to prove the boundedness of several known determinantal inequalities. Additionally, we address the optimization problem of finding the supremum of such ratios over all positive definite matrices, formulating it as a linear optimization program. Finally, for completeness, we include the proofs of theorems that appear to have been previously known but lack accessible proofs.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 525-550"},"PeriodicalIF":1.0,"publicationDate":"2024-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143165106","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 local dimensions of solutions of Brent equations","authors":"Xin Li ,&nbsp;Yixin Bao ,&nbsp;Liping Zhang","doi":"10.1016/j.laa.2024.12.011","DOIUrl":"10.1016/j.laa.2024.12.011","url":null,"abstract":"<div><div>Let <span><math><mo>〈</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>〉</mo></math></span> be the matrix multiplication tensor. The solution set of Brent equations corresponds to the tensor decompositions of <span><math><mo>〈</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>〉</mo></math></span>. We study the local dimensions of solutions of the Brent equations over the field of complex numbers. The rank of Jacobian matrix of Brent equations provides an upper bound of the local dimension, which is well-known. We calculate the ranks for some typical known solutions, which are provided in the databases <span><span>[16]</span></span> and <span><span>[17]</span></span>. We show that the automorphism group of the natural algorithm computing <span><math><mo>〈</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>〉</mo></math></span> is <span><math><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>×</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>×</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo><mo>⋊</mo><mi>Q</mi><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span>, <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> are groups of generalized permutation matrices, <span><math><mi>Q</mi><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span> is a subgroup of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> depending on <em>m</em>, <em>n</em> and <em>p</em>. For other algorithms computing <span><math><mo>〈</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>〉</mo></math></span>, some conditions are given, which imply the corresponding automorphism groups are isomorphic to subgroups of <span><math><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>×</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>×</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo><mo>⋊</mo><mi>Q</mi><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span>. So under these conditions, <span><math><msup><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mi>m</mi><mo>−</mo><mi>n</mi><mo>−</mo><mi>p</mi><mo>−</mo><mn>3</mn></math></span> is a lower bound for the local dimensions of solutions of Brent equations. Moreover, the gap between the lower and upper bounds is discussed.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 489-512"},"PeriodicalIF":1.0,"publicationDate":"2024-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143165104","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 maximal ranks of some third-order quaternion tensors","authors":"Y.G. Liang,&nbsp;Yang Zhang","doi":"10.1016/j.laa.2024.12.010","DOIUrl":"10.1016/j.laa.2024.12.010","url":null,"abstract":"<div><div>A complex tensor <em>T</em> can be considered as a quaternion tensor. Consequently, decomposing <em>T</em> using simple quaternion tensors, rather than simple complex tensors, can potentially result in decompositions with a smaller rank. In this paper, we first present an example demonstrating this. Furthermore, we show that the maximal rank of a 3 × 3 × 3 quaternion tensor is 5, and in doing so provide explicit decompositions into simple tensors with several cases. Finally, we provide the maximal ranks for all <span><math><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>×</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>×</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> quaternion tensors with <span><math><mn>2</mn><mo>≤</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>≤</mo><mn>3</mn></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 405-428"},"PeriodicalIF":1.0,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143165099","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}