Linear Algebra and its Applications最新文献

{"title":"Irreducible matrix representations of quaternions","authors":"Yu Chen","doi":"10.1016/j.laa.2024.11.010","DOIUrl":"10.1016/j.laa.2024.11.010","url":null,"abstract":"<div><div>We determine all irreducible real and complex matrix representations of quaternions and classify them up to equivalence. More over, we show that there is a one-to-one correspondence between the equivalence classes of the irreducible matrix representations and those of the field homomorphisms from the real numbers to the complex numbers.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"706 ","pages":"Pages 55-69"},"PeriodicalIF":1.0,"publicationDate":"2024-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142699044","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":"Jordan embeddings and linear rank preservers of structural matrix algebras","authors":"Ilja Gogić,&nbsp;Mateo Tomašević","doi":"10.1016/j.laa.2024.11.013","DOIUrl":"10.1016/j.laa.2024.11.013","url":null,"abstract":"<div><div>We consider subalgebras <span><math><mi>A</mi></math></span> of the algebra <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> complex matrices that contain all diagonal matrices, known in the literature as the structural matrix algebras (SMAs).</div><div>Let <span><math><mi>A</mi><mo>⊆</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> be an arbitrary SMA. We first show that any commuting family of diagonalizable matrices in <span><math><mi>A</mi></math></span> can be intrinsically simultaneously diagonalized (i.e. the corresponding similarity can be chosen from <span><math><mi>A</mi></math></span>). Using this, we then characterize when one SMA Jordan-embeds into another and in that case we describe the form of such Jordan embeddings. As a consequence, we obtain a description of Jordan automorphisms of SMAs, generalizing Coelho's result on their algebra automorphisms. Next, motivated by the results of Marcus-Moyls and Molnar-Šemrl, connecting the linear rank-one preservers with Jordan embeddings <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>→</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>→</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> (where <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is the algebra of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> upper-triangular matrices) respectively, we show that any linear unital rank-one preserver <span><math><mi>A</mi><mo>→</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is necessarily a Jordan embedding. As the converse fails in general, we also provide a necessary and sufficient condition for when it does hold true. Finally, we obtain a complete description of linear rank preservers <span><math><mi>A</mi><mo>→</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, as maps of the form <span><math><mi>X</mi><mo>↦</mo><mi>S</mi><mrow><mo>(</mo><mi>P</mi><mi>X</mi><mo>+</mo><mo>(</mo><mi>I</mi><mo>−</mo><mi>P</mi><mo>)</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>)</mo></mrow><mi>T</mi></math></span>, for some invertible matrices <span><math><mi>S</mi><mo>,</mo><mi>T</mi><mo>∈</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and a central idempotent <span><math><mi>P</mi><mo>∈</mo><mi>A</mi></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"707 ","pages":"Pages 1-48"},"PeriodicalIF":1.0,"publicationDate":"2024-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142707119","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":"Matrix diagonalisation in sesquilinear symplectic spaces","authors":"Tanvi Jain ,&nbsp;Kirti Kajla","doi":"10.1016/j.laa.2024.11.007","DOIUrl":"10.1016/j.laa.2024.11.007","url":null,"abstract":"<div><div>The symplectic inner product on <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup></math></span> is the sesquilinear form given by<span><span><span><math><mo>[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>]</mo><mo>=</mo><mo>〈</mo><mi>x</mi><mo>,</mo><msub><mrow><mi>J</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub><mi>y</mi><mo>〉</mo><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mi>J</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub></math></span> is the real skew-symmetric, orthogonal <span><math><mn>2</mn><mo>×</mo><mn>2</mn></math></span> block matrix <span><math><mo>[</mo><mtable><mtr><mtd><mn>0</mn></mtd><mtd><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub></mtd></mtr><mtr><mtd><mo>−</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mo>]</mo></math></span>. We derive results analogous to the spectral theorem and singular value decomposition for complex matrices such as Hamiltonian and <em>J</em>-normal matrices, in the sesquilinear symplectic inner product spaces.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"706 ","pages":"Pages 1-23"},"PeriodicalIF":1.0,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698308","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":"Eigenvalues of Toeplitz matrices emerging from finite differences for certain ordinary differential operators","authors":"M. Bogoya ,&nbsp;A. Böttcher ,&nbsp;S.M. Grudsky","doi":"10.1016/j.laa.2024.11.008","DOIUrl":"10.1016/j.laa.2024.11.008","url":null,"abstract":"<div><div>We consider Hermitian Toeplitz matrices emerging from finite linear combinations with non-negative coefficients of the differential operators <span><math><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>k</mi></mrow></msup><msup><mrow><mi>d</mi></mrow><mrow><mn>2</mn><mi>k</mi></mrow></msup><mo>/</mo><mi>d</mi><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn><mi>k</mi></mrow></msup></math></span> over the interval <span><math><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> after discretizing them on a uniform grid of step size <span><math><mn>1</mn><mo>/</mo><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>. The collective distribution in the Szegő–Weyl sense of the eigenvalues of these matrices as <em>n</em> goes to infinity can be described by GLT theory. However, we focus on the asymptotic behavior of the individual eigenvalues, on both the inner eigenvalues in the bulk and on the extreme eigenvalues. The difficulty of the problem is that not only the order of the matrices depends on <em>n</em> but also their so-called symbols. Our main results are third order asymptotic formulas for the eigenvalues in the case <span><math><mi>k</mi><mo>⩽</mo><mn>2</mn></math></span>. These results reveal some basic phenomena one should expect when considering the problem in full generality.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"706 ","pages":"Pages 24-54"},"PeriodicalIF":1.0,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698307","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":"Computing eigenvalues for products of two classes of sign regular matrices to high relative accuracy","authors":"Xiaoxiao Ma ,&nbsp;Yingqing Xiao ,&nbsp;Zhao Yang","doi":"10.1016/j.laa.2024.11.006","DOIUrl":"10.1016/j.laa.2024.11.006","url":null,"abstract":"<div><div>In this paper, we consider how to accurately solve the product eigenvalue problem for the class of sign regular (SR) matrices with signature <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mn>1</mn><mo>,</mo><mo>−</mo><mn>1</mn><mo>)</mo></math></span> and the class of totally nonnegative (TN) matrices, which tend to be extremely ill-conditioned. We present algorithms with <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></math></span> complexity to accurately compute the parameter matrices of products of TN matrices and SR matrices with signature <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>1</mn><mo>,</mo><mo>−</mo><mn>1</mn><mo>)</mo></math></span>. Based on the accurate parameter matrices, all eigenvalues of the product matrix are computed to high relative accuracy. Numerical experiments are provided to confirm the claimed high relative accuracy.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"707 ","pages":"Pages 80-106"},"PeriodicalIF":1.0,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142706748","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":"Comprehensive classification of the algebra generated by two idempotent matrices","authors":"Rounak Biswas,&nbsp;Falguni Roy","doi":"10.1016/j.laa.2024.11.005","DOIUrl":"10.1016/j.laa.2024.11.005","url":null,"abstract":"<div><div>For two idempotent matrix <span><math><mi>P</mi><mo>,</mo><mi>Q</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msup></math></span>, let alg<span><math><mo>(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>P</mi><mo>,</mo><mi>Q</mi><mo>)</mo></math></span> denote the smallest subalgebra of <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msup></math></span> that contains <span><math><mi>P</mi><mo>,</mo><mi>Q</mi></math></span> and the identity matrix <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. This paper provides a complete classification of alg<span><math><mo>(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>P</mi><mo>,</mo><mi>Q</mi><mo>)</mo></math></span> without imposing any restrictions on <em>P</em> and <em>Q</em>. As a result of this classification, the issue of group invertibility within alg<span><math><mo>(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>P</mi><mo>,</mo><mi>Q</mi><mo>)</mo></math></span> is fully resolved.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"705 ","pages":"Pages 185-206"},"PeriodicalIF":1.0,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142653347","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":"Quantum subspace controllability implying full controllability","authors":"Francesca Albertini ,&nbsp;Domenico D'Alessandro","doi":"10.1016/j.laa.2024.11.002","DOIUrl":"10.1016/j.laa.2024.11.002","url":null,"abstract":"<div><div>In the analysis of controllability of finite dimensional quantum systems, <em>subspace controllability</em> refers to the situation where the underlying Hilbert space splits into the direct sum of invariant subspaces, and, on each of such invariant subspaces, it is possible to generate any arbitrary unitary operation using appropriate control functions. This is a typical situation in the presence of symmetries for the dynamics.</div><div>We investigate whether and when if subspace controllability is verified, the addition of an extra Hamiltonian to the dynamics implies full controllability of the system. Under the natural (and necessary) condition that the new Hamiltonian connects all the invariant subspaces, we show that this is always the case, except for a very specific case we shall describe. Even in this specific case, a weaker notion of controllability, controllability of the state (<em>Pure State Controllability</em>) is verified.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"705 ","pages":"Pages 207-229"},"PeriodicalIF":1.0,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142653348","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":"Combinatorial reduction of set functions and matroid permutations through minor invertible product assignment","authors":"Mario Angelelli","doi":"10.1016/j.laa.2024.11.004","DOIUrl":"10.1016/j.laa.2024.11.004","url":null,"abstract":"<div><div>We introduce an algebraic model, based on the determinantal expansion of the product of two matrices, to test combinatorial reductions of set functions. Each term of the determinantal expansion is deformed through a monomial factor in <em>d</em> indeterminates, whose exponents define a <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>-valued set function. By combining the Grassmann-Plücker relations for the two matrices, we derive a family of sparse polynomials, whose factorisation properties in a Laurent polynomial ring are studied and related to information-theoretic notions.</div><div>Under a given genericity condition, we prove the equivalence between combinatorial reductions and determinantal expansions with invertible minor products; specifically, a deformation returns a determinantal expansion if and only if it is induced by a diagonal matrix of units in <span><math><mi>C</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> acting as a kernel in the original determinant expression. This characterisation supports the definition of a new method for checking and recovering combinatorial reductions for matroid permutations.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"705 ","pages":"Pages 89-128"},"PeriodicalIF":1.0,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142653344","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":"Collocation methods for nonlinear differential equations on low-rank manifolds","authors":"Alec Dektor","doi":"10.1016/j.laa.2024.11.001","DOIUrl":"10.1016/j.laa.2024.11.001","url":null,"abstract":"<div><div>We introduce new methods for integrating nonlinear differential equations on low-rank manifolds. These methods rely on interpolatory projections onto the tangent space, enabling low-rank time integration of vector fields that can be evaluated entry-wise. A key advantage of our approach is that it does not require the vector field to exhibit low-rank structure, thereby overcoming significant limitations of traditional dynamical low-rank methods based on orthogonal projection. To construct the interpolatory projectors, we develop a sparse tensor sampling algorithm based on the discrete empirical interpolation method (DEIM) that parameterizes tensor train manifolds and their tangent spaces with cross interpolation. Using these projectors, we propose two time integration schemes on low-rank tensor train manifolds. The first scheme integrates the solution at selected interpolation indices and constructs the solution with cross interpolation. The second scheme generalizes the well-known orthogonal projector-splitting integrator to interpolatory projectors. We demonstrate the proposed methods with applications to several tensor differential equations arising from the discretization of partial differential equations.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"705 ","pages":"Pages 143-184"},"PeriodicalIF":1.0,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142653346","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":"Digraphs with few distinct eigenvalues","authors":"M. Cavers ,&nbsp;B. Miraftab","doi":"10.1016/j.laa.2024.10.028","DOIUrl":"10.1016/j.laa.2024.10.028","url":null,"abstract":"<div><div>This paper provides insight into the problem of characterizing digraphs (with loops permitted) that have few distinct adjacency eigenvalues, or equivalently, characterizing square <span><math><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-matrices that have few distinct eigenvalues. A spectral characterization of strongly connected digraphs whose adjacency matrix has exactly two distinct eigenvalues is given and constructions of such digraphs are described. In addition, bipartite digraphs with exactly three distinct eigenvalues are discussed.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"705 ","pages":"Pages 129-142"},"PeriodicalIF":1.0,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142653345","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}