Linear Algebra and its Applications最新文献

{"title":"Separation of the orbits in representations of SO2 and O2 over R and C","authors":"Martin Jalard","doi":"10.1016/j.laa.2025.04.016","DOIUrl":"10.1016/j.laa.2025.04.016","url":null,"abstract":"<div><div>I provide a minimal set of invariant polynomials separating the orbits for representations of <span><math><msub><mrow><mi>SO</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>O</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> over <span><math><mi>C</mi></math></span> and <span><math><mi>R</mi></math></span>. The idea is to select only polynomials of support of size 2 for <span><math><msub><mrow><mi>SO</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and 4 for <span><math><msub><mrow><mi>O</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. I thus obtain cardinalities in respectively <span><math><mi>O</mi><mo>(</mo><mi>dim</mi><mo>⁡</mo><msup><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> and <span><math><mi>O</mi><mo>(</mo><mi>dim</mi><mo>⁡</mo><msup><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow><mrow><mn>4</mn></mrow></msup><mo>)</mo></math></span>. These cardinalities are much smaller than for generating sets, which require polynomials of arbitrary large supports. Yet a separating set is sufficient for most of the applications. It appears also that real separating sets are smaller than the complex ones, which helps significantly for applications over <span><math><mi>R</mi></math></span>. I finally use the obtained separating set to stratify the real representations by isotropy classes.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"722 ","pages":"Pages 38-66"},"PeriodicalIF":1.0,"publicationDate":"2025-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144070121","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":"Numerical ranges are shadows","authors":"Alan Wiggins ,&nbsp;Edwin Xie","doi":"10.1016/j.laa.2025.05.005","DOIUrl":"10.1016/j.laa.2025.05.005","url":null,"abstract":"<div><div>We present a new perspective on the numerical range of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices as varying “shadows” of an embedding of <span><math><mi>C</mi><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>. This framework gives us geometric proofs of the elliptic range theorem and the Toeplitz-Hausdorff theorem. We apply this perspective to the Berezin Range or linear operators on finite-dimensional supbspaces of the Hardy-Hilbert space <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>D</mi><mo>)</mo></math></span> of the open unit disk <span><math><mi>D</mi></math></span>. We characterize the convexity of the Berezin range for two-dimensional subspaces and show that uncountably many unitary conjugates of a given operator are needed to “cover” the numerical range using Berezin ranges if the boundary of the numerical range is not smooth.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"722 ","pages":"Pages 81-100"},"PeriodicalIF":1.0,"publicationDate":"2025-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144088676","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":"Maxima of the Q-index: Forbidden rainbow Hamilton paths, matchings and linear forests","authors":"Xinye Zhang ,&nbsp;Yongtao Li ,&nbsp;Lihua Feng ,&nbsp;Weijun Liu","doi":"10.1016/j.laa.2025.05.004","DOIUrl":"10.1016/j.laa.2025.05.004","url":null,"abstract":"&lt;div&gt;&lt;div&gt;A classical result in graph theory, proved by Ore and Bondy independently, states that every graph on &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; vertices with more than &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; edges contains a path of length &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;. Another well-known result due to Erdős and Gallai asserts that for &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, every &lt;em&gt;n&lt;/em&gt;-vertex graph with more than &lt;span&gt;&lt;math&gt;&lt;mi&gt;max&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; edges contains a matching of size &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;. In this paper, we establish several spectral analogues for graphs to admit a rainbow substructure. Let &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; be a collection of (not necessarily distinct) graphs on the same vertex set &lt;span&gt;&lt;math&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; be the spectral radius of the signless Laplacian matrix of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;. Firstly, we prove that if &lt;span&gt;&lt;math&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; for every &lt;span&gt;&lt;math&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, then &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; admit a rainbow Hamilton path, i.e., a Hamilton path whose edges come from different &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;'s, unless &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;⋯&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"720 ","pages":"Pages 213-244"},"PeriodicalIF":1.0,"publicationDate":"2025-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143937718","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":"Constructing position vectors for the intersection of two and three linear varieties","authors":"M.A. Facas Vicente ,&nbsp;José Vitória","doi":"10.1016/j.laa.2025.05.003","DOIUrl":"10.1016/j.laa.2025.05.003","url":null,"abstract":"<div><div>Position vectors are much useful in several fields, such as Differential Geometry, Mechanics and in Engineering, in particular in Dimensional Metrology. We generalize, for linear varieties in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, the corresponding results known for the Euclidean ordinary space. The Moore-Penrose inverse of matrices plays an important rôle in this paper. Generalizations for three linear varieties of the Anderson-Duffin formulae are presented. We establish several formulae for a position vector of the intersection of linear varieties. Some characterization of the position vector is provided in terms of centres of spheres. Results, in the context of commuting projections, are given as well.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"720 ","pages":"Pages 373-392"},"PeriodicalIF":1.0,"publicationDate":"2025-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144069470","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":"A connection between Schur and Dieudonné's theorems on spaces of bounded rank matrices","authors":"Clément de Seguins Pazzis","doi":"10.1016/j.laa.2025.05.001","DOIUrl":"10.1016/j.laa.2025.05.001","url":null,"abstract":"<div><div>We use a double-duality argument to give a new proof of Dieudonné's theorem on spaces of singular matrices. The argument connects the situation to the structure of spaces of operators with rank at most 1, and works best over algebraically closed fields.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"720 ","pages":"Pages 393-403"},"PeriodicalIF":1.0,"publicationDate":"2025-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144069466","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":"Two classes of minimal generic fundamental invariants for tensors","authors":"Xin Li ,&nbsp;Liping Zhang ,&nbsp;Hanchen Xia","doi":"10.1016/j.laa.2025.04.028","DOIUrl":"10.1016/j.laa.2025.04.028","url":null,"abstract":"<div><div>Motivated by the problems raised by Bürgisser and Ikenmeyer in <span><span>[16]</span></span>, we discuss two classes of minimal generic fundamental invariants for tensors of order 3. The first one is defined on <span><math><msup><mrow><mo>⊗</mo></mrow><mrow><mn>3</mn></mrow></msup><msup><mrow><mi>C</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span>, where <span><math><mi>m</mi><mo>=</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mn>1</mn></math></span>. We study its construction by obstruction design introduced by Bürgisser and Ikenmeyer, which partially answers one problem raised by them. The second one is defined on <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>ℓ</mi><mi>m</mi></mrow></msup><mo>⊗</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>m</mi><mi>n</mi></mrow></msup><mo>⊗</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>n</mi><mi>ℓ</mi></mrow></msup></math></span>. We study its evaluation on the matrix multiplication tensor <span><math><mo>〈</mo><mi>ℓ</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>〉</mo></math></span> and unit tensor <span><math><mo>〈</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>〉</mo></math></span> when <span><math><mi>ℓ</mi><mo>=</mo><mi>m</mi><mo>=</mo><mi>n</mi></math></span>. The evaluation on the unit tensor leads to the definition of Latin cube and 3-dimensional Alon-Tarsi problem. We generalize some results on Latin square to Latin cube, which enrich the understanding of 3-dimensional Alon-Tarsi problem. It is also natural to generalize the constructions to tensors of other orders. We illustrate the distinction between even and odd dimensional generalizations by concrete examples. Finally, some open problems in related fields are raised.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"720 ","pages":"Pages 174-212"},"PeriodicalIF":1.0,"publicationDate":"2025-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143937717","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":"Highly symmetric lines","authors":"Mikhail Ganzhinov","doi":"10.1016/j.laa.2025.05.002","DOIUrl":"10.1016/j.laa.2025.05.002","url":null,"abstract":"<div><div>A generalization of highly symmetric frames is presented by considering also projective stabilizers of frame vectors. This allows construction of highly symmetric line systems and study of highly symmetric frames in a more unified manner. Construction of highly symmetric line systems involves computation of twisted spherical functions associated with finite groups. Further generalizations include definition of highly symmetric systems of subspaces. We give several examples which illustrate our approach including 3 new kissing configurations which improve lower bounds on the kissing number in <span><math><mi>d</mi><mo>=</mo><mn>10</mn><mo>,</mo><mn>11</mn><mo>,</mo><mn>14</mn></math></span> to 510, 592 and 1932 respectively.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"722 ","pages":"Pages 12-37"},"PeriodicalIF":1.0,"publicationDate":"2025-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144068193","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":"A new criterion for oriented graphs to be determined by their generalized skew spectrum","authors":"Yiquan Chao ,&nbsp;Wei Wang ,&nbsp;Hao Zhang","doi":"10.1016/j.laa.2025.04.026","DOIUrl":"10.1016/j.laa.2025.04.026","url":null,"abstract":"<div><div>Spectral characterization of graphs is an important topic in spectral graph theory which has been studied extensively by researchers in recent years. The study of oriented graphs, however, has received less attention so far. In Qiu et al. (2021) <span><span>[6]</span></span>, the authors gave an arithmetic criterion for an oriented graph to be determined by its <em>generalized skew spectrum</em> (DGSS for short). More precisely, let Σ be an <em>n</em>-vertex oriented graph with skew adjacency matrix <em>S</em> and <span><math><mi>W</mi><mo>(</mo><mi>Σ</mi><mo>)</mo><mo>=</mo><mo>[</mo><mi>e</mi><mo>,</mo><mi>S</mi><mi>e</mi><mo>,</mo><mo>…</mo><mo>,</mo><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>e</mi><mo>]</mo></math></span> be the <em>walk-matrix</em> of Σ, where <em>e</em> is the all-one vector. A theorem of Qiu et al. <span><span>[6]</span></span> shows that a self-converse oriented graph Σ is DGSS, provided that the Smith normal form of <span><math><mi>W</mi><mo>(</mo><mi>Σ</mi><mo>)</mo></math></span> is <span><math><mrow><mi>diag</mi></mrow><mo>(</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>2</mn><mo>,</mo><mn>2</mn><mi>d</mi><mo>)</mo></math></span>, where <em>d</em> is an odd and square-free integer and the number of 1's appeared in the diagonal is precisely <span><math><mo>⌈</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></math></span>. In this paper, we show that the above square-freeness assumptions on <em>d</em> can actually be removed, which significantly improves upon the above theorem. Our new ingredient is a key intermediate result, which is of independent interest: for a self-converse oriented graphs Σ and an odd prime <em>p</em>, if the rank of <span><math><mi>W</mi><mo>(</mo><mi>Σ</mi><mo>)</mo></math></span> is <span><math><mi>n</mi><mo>−</mo><mn>1</mn></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>, then the kernel of <span><math><mi>W</mi><msup><mrow><mo>(</mo><mi>Σ</mi><mo>)</mo></mrow><mrow><mi>T</mi></mrow></msup></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> is <em>anisotropic</em>, i.e., <span><math><msup><mrow><mi>v</mi></mrow><mrow><mi>T</mi></mrow></msup><mi>v</mi><mo>≠</mo><mn>0</mn></math></span> for any <span><math><mn>0</mn><mo>≠</mo><mi>v</mi><mo>∈</mo><mrow><mi>ker</mi></mrow><mspace></mspace><mi>W</mi><msup><mrow><mo>(</mo><mi>Σ</mi><mo>)</mo></mrow><mrow><mi>T</mi></mrow></msup></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"720 ","pages":"Pages 339-349"},"PeriodicalIF":1.0,"publicationDate":"2025-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144069468","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":"Row completion of polynomial and rational matrices","authors":"Agurtzane Amparan ,&nbsp;Itziar Baragaña ,&nbsp;Silvia Marcaida ,&nbsp;Alicia Roca","doi":"10.1016/j.laa.2025.04.023","DOIUrl":"10.1016/j.laa.2025.04.023","url":null,"abstract":"<div><div>We characterize the existence of a polynomial (rational) matrix when its eigenstructure (complete structural data) and some of its rows are prescribed. For polynomial matrices, this problem was solved in <span><span>[1]</span></span> when the polynomial matrix has the same degree as the prescribed submatrix. In that paper, the following row completion problems were also solved arising when the eigenstructure was partially prescribed, keeping the restriction on the degree: the eigenstructure but the row (column) minimal indices, and the finite and/or infinite structures. Here we remove the restriction on the degree, allowing it to be greater than or equal to that of the submatrix. We also generalize the results to rational matrices. Obviously, the results obtained hold for the corresponding column completion problems.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"720 ","pages":"Pages 109-138"},"PeriodicalIF":1.0,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143937715","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":"Tensor product surfaces and quadratic syzygies","authors":"Matthew Weaver","doi":"10.1016/j.laa.2025.04.020","DOIUrl":"10.1016/j.laa.2025.04.020","url":null,"abstract":"<div><div>For <span><math><mi>U</mi><mo>⊆</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>(</mo><msub><mrow><mi>O</mi></mrow><mrow><msup><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>×</mo><msup><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msup></mrow></msub><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo><mo>)</mo></math></span> a four-dimensional vector space, a basis <span><math><mo>{</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>}</mo></math></span> of <em>U</em> defines a rational map <span><math><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>U</mi></mrow></msub><mo>:</mo><mspace></mspace><msup><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>×</mo><msup><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>⇢</mo><msup><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. The tensor product surface associated to <em>U</em> is the closed image <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>U</mi></mrow></msub></math></span> of the map <span><math><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>U</mi></mrow></msub></math></span>. These surfaces arise within the field of geometric modelling, in which case it is particularly desirable to obtain the implicit equation of <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>U</mi></mrow></msub></math></span>. In this paper, we study <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>U</mi></mrow></msub></math></span> via the syzygies of the associated bigraded ideal <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>U</mi></mrow></msub><mo>=</mo><mo>(</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></math></span> when <em>U</em> is free of basepoints, i.e. <span><math><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>U</mi></mrow></msub></math></span> is regular. Expanding upon work of Duarte and Schenck <span><span>[12]</span></span> for such ideals with a linear syzygy, we address the case that <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>U</mi></mrow></msub></math></span> has a quadratic syzygy.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"720 ","pages":"Pages 350-372"},"PeriodicalIF":1.0,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144069469","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}