Marina Arav, Frank J. Hall, Hein van der Holst, Zhongshan Li, Aram Mathivanan, Jiamin Pan, Hanfei Xu, Zheng Yang
{"title":"流形横交相似性研究进展","authors":"Marina Arav, Frank J. Hall, Hein van der Holst, Zhongshan Li, Aram Mathivanan, Jiamin Pan, Hanfei Xu, Zheng Yang","doi":"10.1016/j.laa.2024.05.013","DOIUrl":null,"url":null,"abstract":"Let be an real matrix. As shown in the recent paper S.M. Fallat, H.T. Hall, J.C.-H. Lin, and B.L. Shader (2022) , if the manifolds and (consisting of all real matrices having the same sign pattern as ), both considered as embedded submanifolds of , intersect transversally at , then every superpattern of sgn() also allows a matrix similar to . Those authors introduced a condition on (in terms of certain linear matrix equations) equivalent to the above transversality, called the nonsymmetric strong spectral property (nSSP). In this paper, this transversality property of is characterized using an alternative, more direct and convenient condition, called the similarity-transversality property (STP). Let be a generic matrix of order whose entries are independent variables. The STP of is defined as the full row rank property of the Jacobian matrix of the entries of at the zero entry positions of with respect to the nondiagonal entries of . This new approach makes it possible to take better advantage of the combinatorial structure of the matrix , and provides theoretical foundation for constructing matrices similar to a given matrix while the entries have certain desired signs. In particular, several important classes of zero-nonzero patterns and sign patterns that require or allow this transversality property are identified. Examples illustrating many possible applications (such as diagonalizability, number of distinct eigenvalues, nilpotence, idempotence, semi-stability, eigenvalues and their algebraic and geometric multiplicities, Jordan canonical form, minimal polynomial, and rank) are provided.","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Advances on similarity via transversal intersection of manifolds\",\"authors\":\"Marina Arav, Frank J. Hall, Hein van der Holst, Zhongshan Li, Aram Mathivanan, Jiamin Pan, Hanfei Xu, Zheng Yang\",\"doi\":\"10.1016/j.laa.2024.05.013\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let be an real matrix. As shown in the recent paper S.M. Fallat, H.T. Hall, J.C.-H. Lin, and B.L. Shader (2022) , if the manifolds and (consisting of all real matrices having the same sign pattern as ), both considered as embedded submanifolds of , intersect transversally at , then every superpattern of sgn() also allows a matrix similar to . Those authors introduced a condition on (in terms of certain linear matrix equations) equivalent to the above transversality, called the nonsymmetric strong spectral property (nSSP). In this paper, this transversality property of is characterized using an alternative, more direct and convenient condition, called the similarity-transversality property (STP). Let be a generic matrix of order whose entries are independent variables. The STP of is defined as the full row rank property of the Jacobian matrix of the entries of at the zero entry positions of with respect to the nondiagonal entries of . This new approach makes it possible to take better advantage of the combinatorial structure of the matrix , and provides theoretical foundation for constructing matrices similar to a given matrix while the entries have certain desired signs. In particular, several important classes of zero-nonzero patterns and sign patterns that require or allow this transversality property are identified. Examples illustrating many possible applications (such as diagonalizability, number of distinct eigenvalues, nilpotence, idempotence, semi-stability, eigenvalues and their algebraic and geometric multiplicities, Jordan canonical form, minimal polynomial, and rank) are provided.\",\"PeriodicalId\":18043,\"journal\":{\"name\":\"Linear Algebra and its Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-05-24\",\"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://doi.org/10.1016/j.laa.2024.05.013\",\"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://doi.org/10.1016/j.laa.2024.05.013","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Advances on similarity via transversal intersection of manifolds
Let be an real matrix. As shown in the recent paper S.M. Fallat, H.T. Hall, J.C.-H. Lin, and B.L. Shader (2022) , if the manifolds and (consisting of all real matrices having the same sign pattern as ), both considered as embedded submanifolds of , intersect transversally at , then every superpattern of sgn() also allows a matrix similar to . Those authors introduced a condition on (in terms of certain linear matrix equations) equivalent to the above transversality, called the nonsymmetric strong spectral property (nSSP). In this paper, this transversality property of is characterized using an alternative, more direct and convenient condition, called the similarity-transversality property (STP). Let be a generic matrix of order whose entries are independent variables. The STP of is defined as the full row rank property of the Jacobian matrix of the entries of at the zero entry positions of with respect to the nondiagonal entries of . This new approach makes it possible to take better advantage of the combinatorial structure of the matrix , and provides theoretical foundation for constructing matrices similar to a given matrix while the entries have certain desired signs. In particular, several important classes of zero-nonzero patterns and sign patterns that require or allow this transversality property are identified. Examples illustrating many possible applications (such as diagonalizability, number of distinct eigenvalues, nilpotence, idempotence, semi-stability, eigenvalues and their algebraic and geometric multiplicities, Jordan canonical form, minimal polynomial, and rank) are provided.
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.