Linear Algebra and its Applications最新文献

{"title":"Vertex partitioning and p-energy of graphs","authors":"Saieed Akbari ,&nbsp;Hitesh Kumar ,&nbsp;Bojan Mohar ,&nbsp;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>&gt;</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>&lt;</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 ,&nbsp;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 ,&nbsp;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 ,&nbsp;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 ,&nbsp;Polona Oblak ,&nbsp;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 ,&nbsp;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}

{"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 <em>ℓ</em>-th roots of unity — this is a generalization of <em>D</em>-optimal designs and Hadamard's maximal determinant problem, which involves ±1 matrices. For general values of <em>ℓ</em>, we give sharpened determinantal upper bounds and constructions of matrices of large determinant. The maximal determinant problem in the cases <span><math><mi>ℓ</mi><mo>=</mo><mn>3</mn></math></span>, <span><math><mi>ℓ</mi><mo>=</mo><mn>4</mn></math></span> is similar to the classical Hadamard maximal determinant problem for matrices with entries ±1, and many techniques can be generalized. For <span><math><mi>ℓ</mi><mo>=</mo><mn>3</mn></math></span> 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 <span><math><mi>n</mi><mo>&lt;</mo><mn>14</mn></math></span>. Additionally, we survey the case <span><math><mi>ℓ</mi><mo>=</mo><mn>4</mn></math></span> 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}

{"title":"Multiplicity lists by diameter: All trees of diameter <7","authors":"Charles R. Johnson ,&nbsp;Benjamin Mudrak ,&nbsp;Carlos M. Saiago","doi":"10.1016/j.laa.2025.05.023","DOIUrl":"10.1016/j.laa.2025.05.023","url":null,"abstract":"<div><div>This paper considers the problem of determining all multiplicity lists occurring among Hermitian matrices whose graph is a given tree from a new perspective. For all trees of diameter &lt;7, it is shown how to generate all possible lists. For diameter 5 and 6, this includes many nonlinear trees. In the process, for diameter 4 (double stars), the first succinct and direct description of all ordered lists, for each instance, is given. Observations of topological relationships are helpful.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"723 ","pages":"Pages 33-57"},"PeriodicalIF":1.0,"publicationDate":"2025-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144242091","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":"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 <span><math><mi>P</mi></math></span> of <span><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></span>, where <em>k</em> is a field, a subset <span><math><mi>N</mi><mo>⊆</mo><msup><mrow><mi>k</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> is a <span><math><mi>P</mi></math></span><em>-testing set</em> if any member of <span><math><mi>P</mi></math></span> that vanishes at all points of <span><math><mi>N</mi></math></span>, vanishes all over <span><math><msup><mrow><mi>k</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>; and we say <span><math><mi>N</mi></math></span> is <em>optimal</em> if it has the smallest cardinality among all <span><math><mi>P</mi></math></span>-testing sets. This is related to Lagrangian interpolation of data on a set <span><math><mi>N</mi></math></span> of nodes using functions from <span><math><mi>P</mi></math></span>. We consider a <em>generic version</em> of this interpolation problem, when <span><math><mi>P</mi></math></span> has a monomial basis <span><math><mi>B</mi></math></span> that we identify with a <em>grid</em> (i.e. a finite subset of <span><math><msubsup><mrow><mi>N</mi></mrow><mrow><mn>0</mn></mrow><mrow><mspace></mspace><mi>n</mi></mrow></msubsup></math></span>), each node is an <em>n</em>-tuple of independent variables and the set of nodes is identified with a grid <span><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></span>. 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 <span><math><mi>B</mi><mo>=</mo><mi>C</mi></math></span> is a <em>σ</em>-lower (or <em>σ</em>-upper) grid, where we say <span><math><mi>B</mi></math></span> is a <em>σ-lower</em> (resp., <em>σ-upper</em>) <em>grid</em> if it is a union of intervals of <span><math><msubsup><mrow><mi>N</mi></mrow><mrow><mn>0</mn></mrow><mrow><mspace></mspace><mi>n</mi></mrow></msubsup></math></span> having <em>σ</em> as common origin (resp., endpoint). We give explicit (optimal) <span><math><mi>P</mi></math></span>-testing sets for spaces having monomial bases determined by <em>σ</em>-lower (or <em>σ</em>-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}