A Generalized Eigenvector–Eigenvalue Identity from the Viewpoint of Exterior Algebra

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.