Pub Date : 2025-11-17DOI: 10.1016/j.laa.2025.11.011
Markus Gabl
Matrix completion results deal with the question of when a partially specified symmetric matrix can be completed to a member of certain matrix cones. Results from positive semidefinite matrix completion and completely positive matrix completion have been successfully applied in optimization to greatly reduce the number of variables in conic optimization problems in the space of symmetric matrices. In this text, we go the other direction and show that we can use tools from conic optimization (more precisely: from copositive optimization) to establish a new completion result that complements the existing literature in two regards: firstly, we consider set-completely positive matrix completion, which generalizes completion with respect to the traditional completely positive matrix cone. Secondly, we consider a specification pattern that is not in the scope of classical results for completely positive matrix completion. Namely, we consider arrow-head specification patterns where the width is equal to one. Our theory is applied to a class of quadratic optimization problems.
{"title":"Conic optimization techniques yield sufficient conditions for set-completely positive matrix completion under arrowhead specification pattern","authors":"Markus Gabl","doi":"10.1016/j.laa.2025.11.011","DOIUrl":"10.1016/j.laa.2025.11.011","url":null,"abstract":"<div><div>Matrix completion results deal with the question of when a partially specified symmetric matrix can be completed to a member of certain matrix cones. Results from positive semidefinite matrix completion and completely positive matrix completion have been successfully applied in optimization to greatly reduce the number of variables in conic optimization problems in the space of symmetric matrices. In this text, we go the other direction and show that we can use tools from conic optimization (more precisely: from copositive optimization) to establish a new completion result that complements the existing literature in two regards: firstly, we consider set-completely positive matrix completion, which generalizes completion with respect to the traditional completely positive matrix cone. Secondly, we consider a specification pattern that is not in the scope of classical results for completely positive matrix completion. Namely, we consider arrow-head specification patterns where the width is equal to one. Our theory is applied to a class of quadratic optimization problems.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"731 ","pages":"Pages 215-251"},"PeriodicalIF":1.1,"publicationDate":"2025-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145577904","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}
Pub Date : 2025-11-17DOI: 10.1016/j.laa.2025.11.012
Susie Lu, John Urschel
Given a directed graph G, the spread of G is the largest distance between any two eigenvalues of its adjacency matrix. In 2022, Breen, Riasanovsky, Tait, and Urschel asked what n-vertex directed graph maximizes spread, and whether this graph is undirected. We prove the more general result that the spread of any non-negative matrix A with is at most , which is tight up to an additive factor and exact when n is a multiple of three. Furthermore, our results show that the matrix with maximum spread is always symmetric.
{"title":"On the maximum spread of non-negative matrices","authors":"Susie Lu, John Urschel","doi":"10.1016/j.laa.2025.11.012","DOIUrl":"10.1016/j.laa.2025.11.012","url":null,"abstract":"<div><div>Given a directed graph <em>G</em>, the spread of <em>G</em> is the largest distance between any two eigenvalues of its adjacency matrix. In 2022, Breen, Riasanovsky, Tait, and Urschel asked what <em>n</em>-vertex directed graph maximizes spread, and whether this graph is undirected. We prove the more general result that the spread of any <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> non-negative matrix <em>A</em> with <span><math><msub><mrow><mo>‖</mo><mi>A</mi><mo>‖</mo></mrow><mrow><mi>max</mi></mrow></msub><mo>≤</mo><mn>1</mn></math></span> is at most <span><math><mn>2</mn><mi>n</mi><mo>/</mo><msqrt><mrow><mn>3</mn></mrow></msqrt></math></span>, which is tight up to an additive factor and exact when <em>n</em> is a multiple of three. Furthermore, our results show that the matrix with maximum spread is always symmetric.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"731 ","pages":"Pages 186-195"},"PeriodicalIF":1.1,"publicationDate":"2025-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145577832","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}
Pub Date : 2025-11-14DOI: 10.1016/j.laa.2025.11.010
Luis Felipe Prieto-Martínez , Javier Rico
We propose and investigate a bi-infinite matrix approach to the multiplication and composition of formal Laurent series. We generalize the concept of Riordan matrix to this bi-infinite context, obtaining matrices that are not necessarily lower triangular and are determined, not by a pair of formal power series, but by a pair of formal Laurent series. We extend the First Fundamental Theorem of Riordan Matrices to this setting, as well as the Toeplitz and Lagrange subgroups, that are subgroups of the classical Riordan group. Finally, as an illustrative example, we apply our approach to derive a classical combinatorial identity that cannot be proved using the techniques related to the classical Riordan group, showing that our generalization is not fruitless.
{"title":"Bi-infinite Riordan matrices: A matricial approach to multiplication and composition of formal Laurent series","authors":"Luis Felipe Prieto-Martínez , Javier Rico","doi":"10.1016/j.laa.2025.11.010","DOIUrl":"10.1016/j.laa.2025.11.010","url":null,"abstract":"<div><div>We propose and investigate a bi-infinite matrix approach to the multiplication and composition of formal Laurent series. We generalize the concept of Riordan matrix to this bi-infinite context, obtaining matrices that are not necessarily lower triangular and are determined, not by a pair of formal power series, but by a pair of formal Laurent series. We extend the First Fundamental Theorem of Riordan Matrices to this setting, as well as the Toeplitz and Lagrange subgroups, that are subgroups of the classical Riordan group. Finally, as an illustrative example, we apply our approach to derive a classical combinatorial identity that cannot be proved using the techniques related to the classical Riordan group, showing that our generalization is not fruitless.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"731 ","pages":"Pages 139-159"},"PeriodicalIF":1.1,"publicationDate":"2025-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145577905","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}
Pub Date : 2025-11-13DOI: 10.1016/j.laa.2025.11.005
Aljaž Zalar , Igor Zobovič
Let L be a linear operator on univariate polynomials of bounded degree, mapping into real symmetric matrices, such that its moment matrix is positive definite. It is known that L admits a finitely atomic positive matrix-valued representing measure μ. Any μ with the smallest sum of the ranks of the matricial masses is called minimal. In this paper, we characterize the existence of a minimal representing measure containing a prescribed atom with prescribed rank of the corresponding mass, thus extending a recent result [1] for the scalar-valued case. As a corollary, we obtain a constructive, linear algebraic proof of the strong truncated Hamburger matrix moment problem [16] in the nonsingular case. The results will be important in the study of the truncated univariate rational matrix moment problem.
{"title":"Matricial Gaussian quadrature rules: Nonsingular case","authors":"Aljaž Zalar , Igor Zobovič","doi":"10.1016/j.laa.2025.11.005","DOIUrl":"10.1016/j.laa.2025.11.005","url":null,"abstract":"<div><div>Let <em>L</em> be a linear operator on univariate polynomials of bounded degree, mapping into real symmetric matrices, such that its moment matrix is positive definite. It is known that <em>L</em> admits a finitely atomic positive matrix-valued representing measure <em>μ</em>. Any <em>μ</em> with the smallest sum of the ranks of the matricial masses is called minimal. In this paper, we characterize the existence of a minimal representing measure containing a prescribed atom with prescribed rank of the corresponding mass, thus extending a recent result <span><span>[1]</span></span> for the scalar-valued case. As a corollary, we obtain a constructive, linear algebraic proof of the strong truncated Hamburger matrix moment problem <span><span>[16]</span></span> in the nonsingular case. The results will be important in the study of the truncated univariate rational matrix moment problem.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"731 ","pages":"Pages 160-185"},"PeriodicalIF":1.1,"publicationDate":"2025-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145577830","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}
Pub Date : 2025-11-11DOI: 10.1016/j.laa.2025.11.008
Yasemin Alp
<div><div>A Riordan array is provided with <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span>. Its horizontal half <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>h</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>,</mo><mi>n</mi><mo>+</mo><mi>k</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span>, vertical half <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>h</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mi>k</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span>, <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span> horizontal half <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>p</mi><mi>n</mi><mo>+</mo><mi>r</mi><mo>,</mo><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo><mi>n</mi><mo>+</mo><mi>k</mi><mo>+</mo><mi>r</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span>, <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span> vertical half <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>p</mi><mi>n</mi><mo>−</mo><mi>k</mi><mo>+</mo><mi>r</mi><mo>,</mo><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo><mi>n</mi><mo>+</mo><mi>r</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span>, skew <span><math><mo>(</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>)</mo></math></span> half <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>h</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mo>(</mo><mi>s</mi><mo>−</mo><mn>2</mn><mo>)</mo><mi>k</mi><mo>+</mo><mi>r</mi><mo>,</mo><mi>n</mi><mo>+</mo><mo>(</mo><mi>s</mi><mo>−</mo><mn>1</mn><mo>)</mo><mi>k</mi><mo>+</mo><mi>r</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span>, and <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>)</mo></math></span> half <span><math><msub><mrow><mi>h</mi></mrow><mrow><mo>(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>)</mo><mi>n</mi><mo>+</mo><mo>(</mo><mi>p</mi><mi>s</mi><mo>−</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo><mi>k</mi><mo>+</mo><mi>r</mi><mo>,</mo><mi>p</mi><mi>n</mi><mo>+</mo><mo>(</mo><mi>p</mi><mi>s</mi><mo>−</mo><mi>p</mi><mo>)</mo><mi>k</mi><mo>+</mo><mi>r</mi></mrow></msub></math></span> have previously been considered. In this paper, we take into consideration the <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>)</mo></math></span> halves of the Riordan arrays by replacing <em>p</em> with <span><math><mi>p</mi><mo>−</mo><mn>1</mn></math></span> and <em>s</em> with <span><math><mfrac><mrow><mi>p</mi><mo>+</mo><mi>q</mi><mo>−</m
{"title":"Some applications of the (p,r,s) halves of the Riordan arrays","authors":"Yasemin Alp","doi":"10.1016/j.laa.2025.11.008","DOIUrl":"10.1016/j.laa.2025.11.008","url":null,"abstract":"<div><div>A Riordan array is provided with <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span>. Its horizontal half <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>h</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>,</mo><mi>n</mi><mo>+</mo><mi>k</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span>, vertical half <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>h</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mi>k</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span>, <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span> horizontal half <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>p</mi><mi>n</mi><mo>+</mo><mi>r</mi><mo>,</mo><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo><mi>n</mi><mo>+</mo><mi>k</mi><mo>+</mo><mi>r</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span>, <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span> vertical half <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>p</mi><mi>n</mi><mo>−</mo><mi>k</mi><mo>+</mo><mi>r</mi><mo>,</mo><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo><mi>n</mi><mo>+</mo><mi>r</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span>, skew <span><math><mo>(</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>)</mo></math></span> half <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>h</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mo>(</mo><mi>s</mi><mo>−</mo><mn>2</mn><mo>)</mo><mi>k</mi><mo>+</mo><mi>r</mi><mo>,</mo><mi>n</mi><mo>+</mo><mo>(</mo><mi>s</mi><mo>−</mo><mn>1</mn><mo>)</mo><mi>k</mi><mo>+</mo><mi>r</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span>, and <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>)</mo></math></span> half <span><math><msub><mrow><mi>h</mi></mrow><mrow><mo>(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>)</mo><mi>n</mi><mo>+</mo><mo>(</mo><mi>p</mi><mi>s</mi><mo>−</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo><mi>k</mi><mo>+</mo><mi>r</mi><mo>,</mo><mi>p</mi><mi>n</mi><mo>+</mo><mo>(</mo><mi>p</mi><mi>s</mi><mo>−</mo><mi>p</mi><mo>)</mo><mi>k</mi><mo>+</mo><mi>r</mi></mrow></msub></math></span> have previously been considered. In this paper, we take into consideration the <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>)</mo></math></span> halves of the Riordan arrays by replacing <em>p</em> with <span><math><mi>p</mi><mo>−</mo><mn>1</mn></math></span> and <em>s</em> with <span><math><mfrac><mrow><mi>p</mi><mo>+</mo><mi>q</mi><mo>−</m","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"731 ","pages":"Pages 41-58"},"PeriodicalIF":1.1,"publicationDate":"2025-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145527587","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}
Pub Date : 2025-11-11DOI: 10.1016/j.laa.2025.11.009
K. Sharma, S.K. Panda
Let G be the underlying graph of a real symmetric matrix A. Denote by the principal submatrix of A obtained by deleting the jth row and column, and let denote the algebraic multiplicity of the eigenvalue of A. An index j is called a P-vertex of A if . A graph G on n vertices is said to have property (P) if there exists a nonsingular symmetric matrix A whose underlying graph is G such that every vertex of A is a P-vertex. This work develops a graph-theoretic framework for studying property (P), with particular emphasis on graphs that admit a perfect matching. We analyze bipartite graphs that satisfy property (P) and show that the existence of a perfect matching plays a decisive role in their characterization. In particular, we prove that a tree possesses property (P) if and only if it admits a unique perfect matching, and we present an alternative characterization of unicyclic graphs satisfying property (P). The analysis is then extended to non-bipartite graphs with a unique perfect matching, where we highlight structural features that influence property (P). Furthermore, we construct a family of graphs on n vertices that do not satisfy property (P), but have a nonsingular matrix for which the number of P-vertices is . We also investigate the behavior of property (P) under certain graph operations, such as the edge-joining of graphs, and show that this operation preserves property (P) under specific conditions. In particular, we establish that if two graphs G and H each satisfy property (P), then the graph obtained by joining them with a single edge also satisfies property (P), and we examine the converse of this result.
{"title":"The P-vertex problem for symmetric matrices whose associated graphs admit perfect matchings","authors":"K. Sharma, S.K. Panda","doi":"10.1016/j.laa.2025.11.009","DOIUrl":"10.1016/j.laa.2025.11.009","url":null,"abstract":"<div><div>Let <em>G</em> be the underlying graph of a real symmetric matrix <em>A</em>. Denote by <span><math><mi>A</mi><mo>(</mo><mi>j</mi><mo>)</mo></math></span> the principal submatrix of <em>A</em> obtained by deleting the <em>j</em>th row and column, and let <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>(</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></math></span> denote the algebraic multiplicity of the eigenvalue <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> of <em>A</em>. An index <em>j</em> is called a P-vertex of <em>A</em> if <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>A</mi><mo>(</mo><mi>j</mi><mo>)</mo></mrow></msub><mo>(</mo><mn>0</mn><mo>)</mo><mo>−</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>1</mn></math></span>. A graph <em>G</em> on <em>n</em> vertices is said to have property (P) if there exists a nonsingular symmetric matrix <em>A</em> whose underlying graph is <em>G</em> such that every vertex of <em>A</em> is a P-vertex. This work develops a graph-theoretic framework for studying property (P), with particular emphasis on graphs that admit a perfect matching. We analyze bipartite graphs that satisfy property (P) and show that the existence of a perfect matching plays a decisive role in their characterization. In particular, we prove that a tree possesses property (P) if and only if it admits a unique perfect matching, and we present an alternative characterization of unicyclic graphs satisfying property (P). The analysis is then extended to non-bipartite graphs with a unique perfect matching, where we highlight structural features that influence property (P). Furthermore, we construct a family of graphs on <em>n</em> vertices that do not satisfy property (P), but have a nonsingular matrix for which the number of P-vertices is <span><math><mi>n</mi><mo>−</mo><mn>1</mn></math></span>. We also investigate the behavior of property (P) under certain graph operations, such as the edge-joining of graphs, and show that this operation preserves property (P) under specific conditions. In particular, we establish that if two graphs <em>G</em> and <em>H</em> each satisfy property (P), then the graph obtained by joining them with a single edge also satisfies property (P), and we examine the converse of this result.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"731 ","pages":"Pages 109-138"},"PeriodicalIF":1.1,"publicationDate":"2025-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145577893","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}
Pub Date : 2025-11-10DOI: 10.1016/j.laa.2025.11.003
Lucio Centrone , Luís Fertunani , Claudemir Fideles
We classify gradings on null-filiform Leibniz algebras up to equivalence over arbitrary fields. Furthermore, we provide a basis for the graded identities and determine a basis of the relatively free algebra. As a consequence, we establish that the ideal of all graded identities of null-filiform Leibniz algebras satisfy the Specht property. Finally, we extend these results to infinite-dimensional analogs of null-filiform Leibniz algebras.
{"title":"Gradings and graded identities of null-filiform Leibniz algebras","authors":"Lucio Centrone , Luís Fertunani , Claudemir Fideles","doi":"10.1016/j.laa.2025.11.003","DOIUrl":"10.1016/j.laa.2025.11.003","url":null,"abstract":"<div><div>We classify gradings on null-filiform Leibniz algebras up to equivalence over arbitrary fields. Furthermore, we provide a basis for the graded identities and determine a basis of the relatively free algebra. As a consequence, we establish that the ideal of all graded identities of null-filiform Leibniz algebras satisfy the Specht property. Finally, we extend these results to infinite-dimensional analogs of null-filiform Leibniz algebras.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"731 ","pages":"Pages 59-89"},"PeriodicalIF":1.1,"publicationDate":"2025-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145525694","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}
Pub Date : 2025-11-10DOI: 10.1016/j.laa.2025.11.007
Projesh Nath Choudhury , Krushnachandra Panigrahy
<div><div>Let <span><math><mi>S</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo></math></span> be an ordered set of <em>n</em> distinct positive integers. The <em>m</em>th-order <em>n</em>-dimensional tensor <span><math><msub><mrow><mi>T</mi></mrow><mrow><mo>[</mo><mi>S</mi><mo>]</mo></mrow></msub><mo>=</mo><mo>(</mo><msub><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>…</mo><msub><mrow><mi>i</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></msub><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>…</mo><msub><mrow><mi>i</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></msub><mo>=</mo><mi>GCD</mi><mo>(</mo><msub><mrow><mi>s</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></msub><mo>)</mo></math></span>, the greatest common divisor (GCD) of <span><math><msub><mrow><mi>s</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><mo>,</mo><mo>…</mo></math></span>, and <span><math><msub><mrow><mi>s</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></msub></math></span> is called the GCD tensor on <em>S</em>. The earliest result on GCD tensors goes back to Smith [<em>Proc. Lond. Math. Soc.</em>, 1976], who computed the determinant of GCD matrix on <span><math><mi>S</mi><mo>=</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span> using the Euler's totient function, followed by Beslin–Ligh [<em>Linear Algebra Appl.</em>, 1989] who showed all GCD matrices are positive definite. In this note, we study the positivity of higher-order tensors in the <em>k</em>-mode product. We show that all GCD tensors are strongly completely positive (CP). We then show that GCD tensors are infinite divisible. In fact, we prove that for every positive real number <em>r</em>, the tensor <span><math><msubsup><mrow><mi>T</mi></mrow><mrow><mo>[</mo><mi>S</mi><mo>]</mo></mrow><mrow><mo>∘</mo><mi>r</mi></mrow></msubsup><mo>=</mo><mo>(</mo><msubsup><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>…</mo><msub><mrow><m
{"title":"Positivity of GCD tensors and their determinants","authors":"Projesh Nath Choudhury , Krushnachandra Panigrahy","doi":"10.1016/j.laa.2025.11.007","DOIUrl":"10.1016/j.laa.2025.11.007","url":null,"abstract":"<div><div>Let <span><math><mi>S</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo></math></span> be an ordered set of <em>n</em> distinct positive integers. The <em>m</em>th-order <em>n</em>-dimensional tensor <span><math><msub><mrow><mi>T</mi></mrow><mrow><mo>[</mo><mi>S</mi><mo>]</mo></mrow></msub><mo>=</mo><mo>(</mo><msub><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>…</mo><msub><mrow><mi>i</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></msub><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>…</mo><msub><mrow><mi>i</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></msub><mo>=</mo><mi>GCD</mi><mo>(</mo><msub><mrow><mi>s</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></msub><mo>)</mo></math></span>, the greatest common divisor (GCD) of <span><math><msub><mrow><mi>s</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><mo>,</mo><mo>…</mo></math></span>, and <span><math><msub><mrow><mi>s</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></msub></math></span> is called the GCD tensor on <em>S</em>. The earliest result on GCD tensors goes back to Smith [<em>Proc. Lond. Math. Soc.</em>, 1976], who computed the determinant of GCD matrix on <span><math><mi>S</mi><mo>=</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span> using the Euler's totient function, followed by Beslin–Ligh [<em>Linear Algebra Appl.</em>, 1989] who showed all GCD matrices are positive definite. In this note, we study the positivity of higher-order tensors in the <em>k</em>-mode product. We show that all GCD tensors are strongly completely positive (CP). We then show that GCD tensors are infinite divisible. In fact, we prove that for every positive real number <em>r</em>, the tensor <span><math><msubsup><mrow><mi>T</mi></mrow><mrow><mo>[</mo><mi>S</mi><mo>]</mo></mrow><mrow><mo>∘</mo><mi>r</mi></mrow></msubsup><mo>=</mo><mo>(</mo><msubsup><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>…</mo><msub><mrow><m","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"731 ","pages":"Pages 22-40"},"PeriodicalIF":1.1,"publicationDate":"2025-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528037","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}
Pub Date : 2025-11-10DOI: 10.1016/j.laa.2025.11.006
John Duncan , Colin M. McGregor
This is a continuation of Crabb et al. (2021) [5] and Duncan and McGregor (2022) [6] and (2024) [7] in which we studied the real space H of Hermitian matrices in with respect to norms on . Here we look further at properties relating to H as a Lie algebra. We introduce the concept of well-embeddedness, and in that context we determine the only simple Lie algebras which can appear in the decomposition of H. Other topics covered include radicals, semisimplicity and Levi factors. Throughout, examples are used to illustrate situations that can arise and properties that are established.
{"title":"Hermitians in matrix algebras with operator norm and associated Lie algebras - II","authors":"John Duncan , Colin M. McGregor","doi":"10.1016/j.laa.2025.11.006","DOIUrl":"10.1016/j.laa.2025.11.006","url":null,"abstract":"<div><div>This is a continuation of Crabb et al. (2021) <span><span>[5]</span></span> and Duncan and McGregor (2022) <span><span>[6]</span></span> and (2024) <span><span>[7]</span></span> in which we studied the real space <em>H</em> of Hermitian matrices in <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math></span> with respect to norms on <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. Here we look further at properties relating to <em>H</em> as a Lie algebra. We introduce the concept of well-embeddedness, and in that context we determine the only simple Lie algebras which can appear in the decomposition of <em>H</em>. Other topics covered include radicals, semisimplicity and Levi factors. Throughout, examples are used to illustrate situations that can arise and properties that are established.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"731 ","pages":"Pages 90-108"},"PeriodicalIF":1.1,"publicationDate":"2025-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528038","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}
Pub Date : 2025-11-07DOI: 10.1016/j.laa.2025.11.002
Lily Adlin, Giovani Thai, Samuel Tiscareno, Ryan Tully-Doyle
By designating vertices with variables, a simple undirected graph can be augmented to have an associated representing rational function in two variables taking the complex bi-upper halfplane to itself. We give relations between representing functions of certain products of such graphs by way of Schur complements. We also study the connection between the structure of the graph and the regularity of the representing function at a boundary singularity.
{"title":"Cauchy transforms of colored graphs in two variables","authors":"Lily Adlin, Giovani Thai, Samuel Tiscareno, Ryan Tully-Doyle","doi":"10.1016/j.laa.2025.11.002","DOIUrl":"10.1016/j.laa.2025.11.002","url":null,"abstract":"<div><div>By designating vertices with variables, a simple undirected graph can be augmented to have an associated representing rational function in two variables taking the complex bi-upper halfplane to itself. We give relations between representing functions of certain products of such graphs by way of Schur complements. We also study the connection between the structure of the graph and the regularity of the representing function at a boundary singularity.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"731 ","pages":"Pages 1-21"},"PeriodicalIF":1.1,"publicationDate":"2025-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145486068","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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