Lucas J. Rusnak , Josephine Reynes , Russell Li , Eric Yan , Justin Yu
{"title":"{±1}矩阵和定向超图的行列式","authors":"Lucas J. Rusnak , Josephine Reynes , Russell Li , Eric Yan , Justin Yu","doi":"10.1016/j.laa.2024.08.013","DOIUrl":null,"url":null,"abstract":"<div><p>The determinants of <span><math><mo>{</mo><mo>±</mo><mn>1</mn><mo>}</mo></math></span>-matrices are calculated via the oriented hypergraphic Laplacian and summing over incidence generalizations of vertex cycle-covers. These cycle-covers are signed and partitioned into families based on their hyperedge containment. Every non-edge-monic family is shown to contribute a net value of 0 to the Laplacian, while each edge-monic family is shown to sum to the absolute value of the determinant of the original incidence matrix. Simple symmetries are identified as well as their relationship to Hadamard's maximum determinant problem. Finally, the entries of the incidence matrix are reclaimed using only the signs of an adjacency-minimal set of cycle-covers from an edge-monic family.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The determinant of {±1}-matrices and oriented hypergraphs\",\"authors\":\"Lucas J. Rusnak , Josephine Reynes , Russell Li , Eric Yan , Justin Yu\",\"doi\":\"10.1016/j.laa.2024.08.013\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>The determinants of <span><math><mo>{</mo><mo>±</mo><mn>1</mn><mo>}</mo></math></span>-matrices are calculated via the oriented hypergraphic Laplacian and summing over incidence generalizations of vertex cycle-covers. These cycle-covers are signed and partitioned into families based on their hyperedge containment. Every non-edge-monic family is shown to contribute a net value of 0 to the Laplacian, while each edge-monic family is shown to sum to the absolute value of the determinant of the original incidence matrix. Simple symmetries are identified as well as their relationship to Hadamard's maximum determinant problem. Finally, the entries of the incidence matrix are reclaimed using only the signs of an adjacency-minimal set of cycle-covers from an edge-monic family.</p></div>\",\"PeriodicalId\":18043,\"journal\":{\"name\":\"Linear Algebra and its Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-08-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Linear Algebra and its Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0024379524003343\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Linear Algebra and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0024379524003343","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
The determinant of {±1}-matrices and oriented hypergraphs
The determinants of -matrices are calculated via the oriented hypergraphic Laplacian and summing over incidence generalizations of vertex cycle-covers. These cycle-covers are signed and partitioned into families based on their hyperedge containment. Every non-edge-monic family is shown to contribute a net value of 0 to the Laplacian, while each edge-monic family is shown to sum to the absolute value of the determinant of the original incidence matrix. Simple symmetries are identified as well as their relationship to Hadamard's maximum determinant problem. Finally, the entries of the incidence matrix are reclaimed using only the signs of an adjacency-minimal set of cycle-covers from an edge-monic family.
期刊介绍：
Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
