Linear Algebra and its Applications最新文献_第6页

Minimum-norm solutions of the non-symmetric semidefinite Procrustes problem 非对称半定Procrustes问题的最小范数解

IF 1 3区数学

Linear Algebra and its Applications Pub Date : 2025-06-16 DOI: 10.1016/j.laa.2025.06.005

Nicolas Gillis, Stefano Sicilia

{"title":"Minimum-norm solutions of the non-symmetric semidefinite Procrustes problem","authors":"Nicolas Gillis, Stefano Sicilia","doi":"10.1016/j.laa.2025.06.005","DOIUrl":"10.1016/j.laa.2025.06.005","url":null,"abstract":"<div><div>Given two matrices <math><mi>X</mi><mo>,</mo><mi>B</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>m</mi></mrow></msup></math> and a set <math><mi>A</mi><mo>⊆</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msup></math>, a Procrustes problem consists in finding a matrix <math><mi>A</mi><mo>∈</mo><mi>A</mi></math> such that the Frobenius norm of <math><mi>A</mi><mi>X</mi><mo>−</mo><mi>B</mi></math> is minimized. When <math><mi>A</mi></math> is the set of the matrices whose symmetric part is positive semidefinite, we obtain the so-called non-symmetric positive semidefinite Procrustes (NSPSDP) problem. The NSPSDP problem arises in the estimation of compliance or stiffness matrix in solid and elastic structures. If X has rank r, Baghel et al. (2022) [4] proposed a three-step semi-analytical approach: (1) construct a reduced NSPSDP problem in dimension <math><mi>r</mi><mo>×</mo><mi>r</mi></math>, (2) solve the reduced problem by means of a fast gradient method with a linear rate of convergence, and (3) post-process the solution of the reduced problem to construct a solution of the larger original NSPSDP problem. In this paper, we revisit this approach of Baghel et al. and identify an unnecessary assumption used by the authors leading to cases where their algorithm cannot attain a minimum and produces solutions with unbounded norm. In fact, revising the post-processing phase of their semi-analytical approach, we show that the infimum of the NSPSDP problem is always attained, and we show how to compute a minimum-norm solution. We also prove that the symmetric part of the computed solution has minimum rank bounded by r, and that the skew-symmetric part has rank bounded by 2r. Several numerical examples show the efficiency of this algorithm, both in terms of computational speed and of finding optimal minimum-norm solutions.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"724 ","pages":"Pages 21-48"},"PeriodicalIF":1.0,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144313378","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}

引用次数: 0

Laplacian eigenvalue distribution in terms of degree sequence 阶序列的拉普拉斯特征值分布

IF 1 3区数学

Linear Algebra and its Applications Pub Date : 2025-06-16 DOI: 10.1016/j.laa.2025.06.010

S. Akbari , M. Alaeiyan , M. Darougheh

引用次数: 0

Vertex partitioning and p-energy of graphs 图的顶点划分与p-能

IF 1 3区数学

Linear Algebra and its Applications Pub Date : 2025-06-16 DOI: 10.1016/j.laa.2025.06.009

Saieed Akbari , Hitesh Kumar , Bojan Mohar , Shivaramakrishna Pragada

引用次数: 0

Davis-Wielandt shells of 4 by 4 matrices 4 × 4矩阵的Davis-Wielandt壳层

IF 1 3区数学

Linear Algebra and its Applications Pub Date : 2025-06-16 DOI: 10.1016/j.laa.2025.06.006

Mao-Ting Chien , Hiroshi Nakazato

引用次数: 0

Orthogonalisability of joins of graphs 图的连接的正交性

IF 1 3区数学

Linear Algebra and its Applications Pub Date : 2025-06-06 DOI: 10.1016/j.laa.2025.06.001

Rupert H. Levene , Polona Oblak , Helena Šmigoc

引用次数: 0

On indefinite-inner-product spaces induced by non-zero-scaled hypercomplex numbers 非零标度超复数诱导的不定内积空间

IF 1 3区数学

Linear Algebra and its Applications Pub Date : 2025-06-06 DOI: 10.1016/j.laa.2025.06.002

Daniel Alpay , Ilwoo Cho

引用次数: 0

Maximal determinants of matrices over the roots of unity 单位根上矩阵的极大行列式

IF 1 3区数学

Linear Algebra and its Applications Pub Date : 2025-06-06 DOI: 10.1016/j.laa.2025.05.024

Guillermo Nuñez Ponasso

{"title":"Maximal determinants of matrices over the roots of unity","authors":"Guillermo Nuñez Ponasso","doi":"10.1016/j.laa.2025.05.024","DOIUrl":"10.1016/j.laa.2025.05.024","url":null,"abstract":"<div><div>We study the maximum absolute value of the determinant of matrices with entries in the set of ℓ-th roots of unity — this is a generalization of D-optimal designs and Hadamard's maximal determinant problem, which involves ±1 matrices. For general values of ℓ, we give sharpened determinantal upper bounds and constructions of matrices of large determinant. The maximal determinant problem in the cases <math><mi>ℓ</mi><mo>=</mo><mn>3</mn></math>, <math><mi>ℓ</mi><mo>=</mo><mn>4</mn></math> is similar to the classical Hadamard maximal determinant problem for matrices with entries ±1, and many techniques can be generalized. For <math><mi>ℓ</mi><mo>=</mo><mn>3</mn></math> we give an additional construction of matrices with large determinant, and calculate the value of the maximal determinant of matrices with entries in the third-roots of unity for all orders <math><mi>n</mi><mo><</mo><mn>14</mn></math>. Additionally, we survey the case <math><mi>ℓ</mi><mo>=</mo><mn>4</mn></math> and exhibit an infinite family of maximal determinant matrices of odd order over the fourth roots of unity.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"723 ","pages":"Pages 201-243"},"PeriodicalIF":1.0,"publicationDate":"2025-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144291012","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}

引用次数: 0

Algebraic notes on testing sets for lower and upper grids 下网格和上网格测试集的代数注释

IF 1 3区数学

Linear Algebra and its Applications Pub Date : 2025-06-03 DOI: 10.1016/j.laa.2025.05.022

Eduardo Marques de Sá

{"title":"Algebraic notes on testing sets for lower and upper grids","authors":"Eduardo Marques de Sá","doi":"10.1016/j.laa.2025.05.022","DOIUrl":"10.1016/j.laa.2025.05.022","url":null,"abstract":"<div><div>For a given finite dimensional subspace <math><mi>P</mi></math> of <math><mi>k</mi><mo>[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></math>, where k is a field, a subset <math><mi>N</mi><mo>⊆</mo><msup><mrow><mi>k</mi></mrow><mrow><mi>n</mi></mrow></msup></math> is a <math><mi>P</mi></math>-testing set if any member of <math><mi>P</mi></math> that vanishes at all points of <math><mi>N</mi></math>, vanishes all over <math><msup><mrow><mi>k</mi></mrow><mrow><mi>n</mi></mrow></msup></math>; and we say <math><mi>N</mi></math> is optimal if it has the smallest cardinality among all <math><mi>P</mi></math>-testing sets. This is related to Lagrangian interpolation of data on a set <math><mi>N</mi></math> of nodes using functions from <math><mi>P</mi></math>. We consider a generic version of this interpolation problem, when <math><mi>P</mi></math> has a monomial basis <math><mi>B</mi></math> that we identify with a grid (i.e. a finite subset of <math><msubsup><mrow><mi>N</mi></mrow><mrow><mn>0</mn></mrow><mrow><mspace></mspace><mi>n</mi></mrow></msubsup></math>), each node is an n-tuple of independent variables and the set of nodes is identified with a grid <math><mi>C</mi><mo>⊆</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mn>0</mn></mrow><mrow><mspace></mspace><mi>n</mi></mrow></msubsup></math>. A corollary to our main result offers an explicit formula for the determinant of the linear system corresponding to the generic interpolation problem in case <math><mi>B</mi><mo>=</mo><mi>C</mi></math> is a σ-lower (or σ-upper) grid, where we say <math><mi>B</mi></math> is a σ-lower (resp., σ-upper) grid if it is a union of intervals of <math><msubsup><mrow><mi>N</mi></mrow><mrow><mn>0</mn></mrow><mrow><mspace></mspace><mi>n</mi></mrow></msubsup></math> having σ as common origin (resp., endpoint). We give explicit (optimal) <math><mi>P</mi></math>-testing sets for spaces having monomial bases determined by σ-lower (or σ-upper) grids. The corollaries at the end, for the finite field case, have potential use in Number Theory and Coding Theory.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"723 ","pages":"Pages 78-98"},"PeriodicalIF":1.0,"publicationDate":"2025-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144255188","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}

引用次数: 0

Grothendieck group of the Leavitt path algebra over power graphs of prime-power cyclic groups 素幂循环群幂图上的Leavitt路径代数的Grothendieck群

IF 1 3区数学

Linear Algebra and its Applications Pub Date : 2025-06-02 DOI: 10.1016/j.laa.2025.05.021

Aslı Güçlükan İlhan , Müge Kanuni , Ekrem Şimşek

引用次数: 0

Decomposable numerical ranges of normal matrices 正规矩阵的可分解数值范围

IF 1 3区数学

Linear Algebra and its Applications Pub Date : 2025-05-23 DOI: 10.1016/j.laa.2025.05.016

Pan-Shun Lau , Chi-Kwong Li , Nung-Sing Sze

引用次数: 0