{"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 <span><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></span> and a set <span><math><mi>A</mi><mo>⊆</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msup></math></span>, a Procrustes problem consists in finding a matrix <span><math><mi>A</mi><mo>∈</mo><mi>A</mi></math></span> such that the Frobenius norm of <span><math><mi>A</mi><mi>X</mi><mo>−</mo><mi>B</mi></math></span> is minimized. When <span><math><mi>A</mi></math></span> 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 <em>X</em> has rank <em>r</em>, Baghel et al. (2022) <span><span>[4]</span></span> proposed a three-step semi-analytical approach: (1) construct a reduced NSPSDP problem in dimension <span><math><mi>r</mi><mo>×</mo><mi>r</mi></math></span>, (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 <em>r</em>, and that the skew-symmetric part has rank bounded by 2<em>r</em>. 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}
{"title":"Banded frames with banded duals","authors":"Kevin Lim , Chengpei Liu , Tim Wertz","doi":"10.1016/j.laa.2025.06.014","DOIUrl":"10.1016/j.laa.2025.06.014","url":null,"abstract":"<div><div>Banded invertible matrices typically do not have banded inverses, but the case when the inverse is banded is characterized by a factorization into block diagonal matrices. In this paper, we extend this result to full rank but non-invertible banded matrices. Such matrices have either a left or right inverse, but not both. These matrices arise naturally in frame theory, where a surjective matrix corresponds to a frame, and its right inverses correspond to dual frames. We generalize a theorem of Asplund and apply it to describe banded matrices with banded left or right inverses. Equivalently, we characterize banded finite frames with banded dual frames.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"724 ","pages":"Pages 120-136"},"PeriodicalIF":1.0,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144365025","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":"Laplacian eigenvalue distribution in terms of degree sequence","authors":"S. Akbari , M. Alaeiyan , M. Darougheh","doi":"10.1016/j.laa.2025.06.010","DOIUrl":"10.1016/j.laa.2025.06.010","url":null,"abstract":"<div><div>Let <em>G</em> be a graph with degree sequence <span><math><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≥</mo><mo>⋯</mo><mo>≥</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. Suppose that <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>I</mi></math></span> denotes the number of Laplacian eigenvalues of <em>G</em> in an interval <em>I</em>. This paper presents some bounds on the number of Laplacian eigenvalues contained in the various subintervals of <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo>]</mo></math></span> in terms of the degree sequence of <em>G</em>. We show that <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>[</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>]</mo><mo>=</mo><mn>2</mn></math></span> if and only if <span><math><mi>G</mi><mo>∈</mo><mo>{</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>}</mo></math></span>. Additionally, we characterize all graphs for which <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>[</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>n</mi><mo>]</mo><mo>=</mo><mn>2</mn></math></span>. Moreover, we classify all graphs such that <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>[</mo><mn>0</mn><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>]</mo><mo>=</mo><mn>2</mn></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"724 ","pages":"Pages 49-61"},"PeriodicalIF":1.0,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144321335","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":"Vertex partitioning and p-energy of graphs","authors":"Saieed Akbari , Hitesh Kumar , Bojan Mohar , Shivaramakrishna Pragada","doi":"10.1016/j.laa.2025.06.009","DOIUrl":"10.1016/j.laa.2025.06.009","url":null,"abstract":"<div><div>For a Hermitian matrix <em>A</em> of order <em>n</em> with eigenvalues <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo><mo>≥</mo><mo>⋯</mo><mo>≥</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, define<span><span><span><math><msubsup><mrow><mi>E</mi></mrow><mrow><mi>p</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><munder><mo>∑</mo><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>></mo><mn>0</mn></mrow></munder><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>(</mo><mi>A</mi><mo>)</mo><mo>,</mo><mspace></mspace><msubsup><mrow><mi>E</mi></mrow><mrow><mi>p</mi></mrow><mrow><mo>−</mo></mrow></msubsup><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><munder><mo>∑</mo><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo><</mo><mn>0</mn></mrow></munder><mo>|</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi></mrow></msup><mo>,</mo></math></span></span></span> to be the positive and the negative <em>p</em>-energy of <em>A</em>, respectively. In this note, first we show that if <span><math><mi>A</mi><mo>=</mo><msubsup><mrow><mo>[</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>]</mo></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>k</mi></mrow></msubsup></math></span>, where <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi><mi>i</mi></mrow></msub></math></span> are square matrices, then<span><span><span><math><msubsup><mrow><mi>E</mi></mrow><mrow><mi>p</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>(</mo><mi>A</mi><mo>)</mo><mo>≥</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>k</mi></mrow></munderover><msubsup><mrow><mi>E</mi></mrow><mrow><mi>p</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi><mi>i</mi></mrow></msub><mo>)</mo><mo>,</mo><mspace></mspace><msubsup><mrow><mi>E</mi></mrow><mrow><mi>p</mi></mrow><mrow><mo>−</mo></mrow></msubsup><mo>(</mo><mi>A</mi><mo>)</mo><mo>≥</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>k</mi></mrow></munderover><msubsup><mrow><mi>E</mi></mrow><mrow><mi>p</mi></mrow><mrow><mo>−</mo></mrow></msubsup><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi><mi>i</mi></mrow></msub><mo>)</mo><mo>,</mo></math></span></span></span> for any real number <span><math><mi>p</mi><mo>≥</mo><mn>1</mn></math></span>. We then apply the previous inequalities to establish lower bounds for <em>p</em>-energy of the adjacency matrix of graphs.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"724 ","pages":"Pages 96-107"},"PeriodicalIF":1.0,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144321276","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":"Davis-Wielandt shells of 4 by 4 matrices","authors":"Mao-Ting Chien , Hiroshi Nakazato","doi":"10.1016/j.laa.2025.06.006","DOIUrl":"10.1016/j.laa.2025.06.006","url":null,"abstract":"<div><div>In this paper, we study possible degrees of the boundary generating surfaces of the Davis-Wielandt shells of 4-by-4 upper triangular unitarily irreducible matrices. The degree can be any even number between 6 and 36 except 14,26 and 30.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"723 ","pages":"Pages 182-200"},"PeriodicalIF":1.0,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144291206","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":"Computing H-equations with 2-by-2 integral matrices","authors":"Gemma Bastardas , Enric Ventura","doi":"10.1016/j.laa.2025.06.007","DOIUrl":"10.1016/j.laa.2025.06.007","url":null,"abstract":"<div><div>We study the transference through finite index extensions of the notion of equational coherence, as well as its effective counterpart. We deduce an explicit algorithm for solving the following algorithmic problem about size two integral invertible matrices: “given <span><math><msub><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>;</mo><mi>g</mi><mo>∈</mo><msub><mrow><mi>PSL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo></math></span>, decide whether <em>g</em> is algebraic over the subgroup <span><math><mi>H</mi><mo>=</mo><mrow><mo>〈</mo><msub><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>〉</mo></mrow><mo>⩽</mo><msub><mrow><mi>PSL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo></math></span> (i.e., whether there exist a non-trivial <em>H</em>-equation <span><math><mi>w</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><mi>H</mi><mo>⁎</mo><mrow><mo>〈</mo><mi>x</mi><mo>〉</mo></mrow></math></span> such that <span><math><mi>w</mi><mo>(</mo><mi>g</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span>) and, in the affirmative case, compute finitely many such <em>H</em>-equations <span><math><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><mi>H</mi><mo>⁎</mo><mrow><mo>〈</mo><mi>x</mi><mo>〉</mo></mrow></math></span> further satisfying that any <span><math><mi>w</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><mi>H</mi><mo>⁎</mo><mrow><mo>〈</mo><mi>x</mi><mo>〉</mo></mrow></math></span> with <span><math><mi>w</mi><mo>(</mo><mi>g</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span> is a product of conjugates of <span><math><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span>”. The same problem for square matrices of size 4 and bigger is unsolvable.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"724 ","pages":"Pages 218-241"},"PeriodicalIF":1.0,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144514392","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":"Leaf as a Poincaré convex domain associated with an endomorphism on a real inner product space","authors":"Hiroyuki Ogawa","doi":"10.1016/j.laa.2025.06.004","DOIUrl":"10.1016/j.laa.2025.06.004","url":null,"abstract":"<div><div>We define a subset of the closure of the upper half plane associated with an endomorphism on a real inner product space, which is called the leaf. When the dimension of the space is at least 3, the leaf is a convex with respect to the Poincaré metric, and contains all eigenvalues with nonnegative imaginary part. Moreover, the leaf of a normal endomorphism is the minimum Poincaré convex domain containing all eigenvalues with nonnegative imaginary part. The most commonly studied convex domain containing eigenvalues is number range. Numerical range is convex with respect to the Euclidean metric on <span><math><mi>C</mi></math></span>, so numerical range has less information than leaf about real eigenvalues. We provide a new visual approach to endomorphisms.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"724 ","pages":"Pages 62-82"},"PeriodicalIF":1.0,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144321336","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 spectral map for weighted Cauchy matrices is an involution","authors":"Alexander Pushnitski , Sergei Treil","doi":"10.1016/j.laa.2025.06.003","DOIUrl":"10.1016/j.laa.2025.06.003","url":null,"abstract":"<div><div>Let <em>N</em> be a natural number. We consider weighted Cauchy matrices of the form<span><span><span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi><mo>,</mo><mi>A</mi></mrow></msub><mo>=</mo><msubsup><mrow><mo>{</mo><mfrac><mrow><msqrt><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>j</mi></mrow></msub><msub><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msqrt></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfrac><mo>}</mo></mrow><mrow><mi>j</mi><mo>,</mo><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>N</mi></mrow></msubsup><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>N</mi></mrow></msub></math></span> are positive real numbers and <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>N</mi></mrow></msub></math></span> are distinct positive real numbers, listed in increasing order. Let <span><math><msub><mrow><mi>b</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>N</mi></mrow></msub></math></span> be the eigenvalues of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi><mo>,</mo><mi>A</mi></mrow></msub></math></span>, listed in increasing order. Let <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> be positive real numbers such that <span><math><msqrt><mrow><msub><mrow><mi>B</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msqrt></math></span> is the Euclidean norm of the orthogonal projection of the vector<span><span><span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>=</mo><mo>(</mo><msqrt><mrow><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msqrt><mo>,</mo><mo>…</mo><mo>,</mo><msqrt><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>N</mi></mrow></msub></mrow></msqrt><mo>)</mo></math></span></span></span> onto the <em>k</em>'th eigenspace of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi><mo>,</mo><mi>A</mi></mrow></msub></math></span>. We prove that the spectral map <span><math><mo>(</mo><mi>a</mi><mo>,</mo><mi>A</mi><mo>)</mo><mo>↦</mo><mo>(</mo><mi>b</mi><mo>,</mo><mi>B</mi><mo>)</mo></math></span> is an involution and discuss simple properties of this map.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"724 ","pages":"Pages 1-11"},"PeriodicalIF":1.0,"publicationDate":"2025-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144313565","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":"Orthogonalisability of joins of graphs","authors":"Rupert H. Levene , Polona Oblak , Helena Šmigoc","doi":"10.1016/j.laa.2025.06.001","DOIUrl":"10.1016/j.laa.2025.06.001","url":null,"abstract":"<div><div>A graph is said to be orthogonalisable if the set of real symmetric matrices whose off-diagonal pattern is prescribed by its edges contains an orthogonal matrix. We determine some necessary and some sufficient conditions on the sizes of the connected components of two graphs for their join to be orthogonalisable. In some cases, those conditions coincide, and we present several families of joins of graphs that are orthogonalisable.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"723 ","pages":"Pages 162-181"},"PeriodicalIF":1.0,"publicationDate":"2025-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144270437","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 indefinite-inner-product spaces induced by non-zero-scaled hypercomplex numbers","authors":"Daniel Alpay , Ilwoo Cho","doi":"10.1016/j.laa.2025.06.002","DOIUrl":"10.1016/j.laa.2025.06.002","url":null,"abstract":"<div><div>In this paper, we consider a new type of adjoint <span><math><mo>[</mo><mo>⁎</mo><mo>]</mo></math></span> on the algebra <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> of all <em>t</em>-scaled hypercomplex numbers over the real field <span><math><mi>R</mi></math></span>, for all “non-zero” scales <span><math><mi>t</mi><mo>∈</mo><mi>R</mi><mo>∖</mo><mrow><mo>{</mo><mn>0</mn><mo>}</mo></mrow></math></span>. We show that such a <span><math><mi>R</mi></math></span>-adjoint <span><math><mo>[</mo><mo>⁎</mo><mo>]</mo></math></span> generates a well-defined indefinite inner product <span><math><msub><mrow><mo>[</mo><mo>,</mo><mo>]</mo></mrow><mrow><mi>t</mi></mrow></msub></math></span> on <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span>, inducing a complete indefinite inner product space <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mrow><mo>(</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>,</mo><msub><mrow><mo>[</mo><mo>,</mo><mo>]</mo></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></mrow></math></span> over <span><math><mi>R</mi></math></span>. Analysis and operator theory on <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> is considered up to this adjoint <span><math><mo>[</mo><mo>⁎</mo><mo>]</mo></math></span>. As application, by regarding <em>t</em>-scaled hypercomplex numbers of <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> as embedded subset <span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>t</mi></mrow></msubsup></math></span> of <span><math><msub><mrow><mi>M</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>C</mi><mo>)</mo></mrow></math></span>, the corresponding (usual operator-theoretic) spectral theory on <span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>t</mi></mrow></msubsup></math></span> is studied (over the complex field <span><math><mi>C</mi></math></span>). And we study relations between these usual spectral-theoretic results and the operator-theoretic results obtained from the <span><math><mo>[</mo><mo>⁎</mo><mo>]</mo></math></span>-depending structures; and then the free distributions of self-adjoint matrices of <span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>t</mi></mrow></msubsup></math></span> are characterized up to the normalized trace <em>τ</em> on <span><math><msub><mrow><mi>M</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>C</mi><mo>)</mo></mrow></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"723 ","pages":"Pages 99-161"},"PeriodicalIF":1.0,"publicationDate":"2025-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144255189","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}
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