{"title":"A Generalized Eigenvector–Eigenvalue Identity from the Viewpoint of Exterior Algebra","authors":"Małgorzata Stawiska","doi":"10.1007/s00006-025-01375-w","DOIUrl":null,"url":null,"abstract":"<div><p>We consider square matrices over <span>\\(\\mathbb {C}\\)</span> satisfying an identity relating their eigenvalues and the corresponding eigenvectors re-proved and discussed by Denton, Parker, Tao and Zhang, called the eigenvector-eigenvalue identity. We prove that for an eigenvalue <span>\\(\\lambda \\)</span> of a given matrix, the identity holds if and only if the geometric multiplicity of <span>\\(\\lambda \\)</span> equals its algebraic multiplicity. We do not make any other assumptions on the matrix and allow the multiplicity of the eigenvalue to be greater than 1, which provides a substantial generalization of the identity. In the proof, we use exterior algebra, particularly the properties of higher adjugates of a matrix.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"35 2","pages":""},"PeriodicalIF":1.1000,"publicationDate":"2025-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Applied Clifford Algebras","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00006-025-01375-w","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
We consider square matrices over \(\mathbb {C}\) satisfying an identity relating their eigenvalues and the corresponding eigenvectors re-proved and discussed by Denton, Parker, Tao and Zhang, called the eigenvector-eigenvalue identity. We prove that for an eigenvalue \(\lambda \) of a given matrix, the identity holds if and only if the geometric multiplicity of \(\lambda \) equals its algebraic multiplicity. We do not make any other assumptions on the matrix and allow the multiplicity of the eigenvalue to be greater than 1, which provides a substantial generalization of the identity. In the proof, we use exterior algebra, particularly the properties of higher adjugates of a matrix.
期刊介绍：
Advances in Applied Clifford Algebras (AACA) publishes high-quality peer-reviewed research papers as well as expository and survey articles in the area of Clifford algebras and their applications to other branches of mathematics, physics, engineering, and related fields. The journal ensures rapid publication and is organized in six sections: Analysis, Differential Geometry and Dirac Operators, Mathematical Structures, Theoretical and Mathematical Physics, Applications, and Book Reviews.
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