A generalized eigenvector-eigenvalue identity from the viewpoint of exterior algebra
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2019
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866909556341211136 |
|---|---|
| author | Stawiska, Malgorzata |
| author_facet | Stawiska, Malgorzata |
| contents | 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 $λ$ of a given matrix the identity holds if and only if the geometric multiplicity of $λ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1912_06967 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | A generalized eigenvector-eigenvalue identity from the viewpoint of exterior algebra Stawiska, Malgorzata Rings and Algebras 15A15, 15A24, 15A75 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 $λ$ of a given matrix the identity holds if and only if the geometric multiplicity of $λ$ 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. |
| title | A generalized eigenvector-eigenvalue identity from the viewpoint of exterior algebra |
| topic | Rings and Algebras 15A15, 15A24, 15A75 |
| url | https://arxiv.org/abs/1912.06967 |