A generalized eigenvector-eigenvalue identity from the viewpoint of exterior algebra
Malgorzata Stawiska
TL;DR
This work generalizes the eigenvector–eigenvalue identity to arbitrary complex matrices by exploiting exterior algebra and higher adjugates. It shows that for an eigenvalue $\lambda$ with geometric multiplicity $k$ (1 ≤ k ≤ n-1), the algebraic multiplicity equals $k$ if and only if $\operatorname{adj}_k(A-\lambda I)$ equals a scalar multiple of a rank-one factor $v w^\top$, where $v$ and $w$ are wedge products of bases of $\ker(A-\lambda I)$ and its dual, scaled by $(-1)^kP^{(k)}(\lambda)/k!$. The method relies on the identities of compound/adjugate matrices and the derivative relation $P^{(k)}(\lambda)=(-1)^k k!\operatorname{tr}(\operatorname{adj}_k(A-\lambda I))$, offering a unified framework that recovers known results for Hermitian/normal matrices as special cases. The results provide new structural insight into how eigenvectors relate to eigenvalues via higher adjugates, with implications for linear algebra theory and potential applications to spectral analysis.
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 $λ$ 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.