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arXiv · 1912.06967

A generalized eigenvector-eigenvalue identity from the viewpoint of exterior algebra

Malgorzata Stawiska
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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.

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Malgorzata Stawiska. 2019-12-15. A generalized eigenvector-eigenvalue identity from the viewpoint of exterior algebra. https://doi.org/10.1007/s00006-025-01375-w

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