Eigenvectors from Eigenvalues: a survey of a basic identity in linear algebra

Theorem 1 (Eigenvector-eigenvalue identity) Let {A} be an {n \times n} Hermitian matrix, with eigenvalues {\lambda_1(A),\dots,\lambda_n(A)}. Let {v_i} be a unit eigenvector corresponding to the eigenvalue {\lambda_i(A)}, and let {v_{i,j}} be the {j^{th}} component of {v_i}. Then

\displaystyle |v_{i,j}|^2 \prod_{k=1; k \neq i}^n (\lambda_i(A) - \lambda_k(A)) = \prod_{k=1}^{n-1} (\lambda_i(A) - \lambda_k(M_j))

where {M_j} is the {n-1 \times n-1} Hermitian matrix formed by deleting the {j^{th}} row and column from {A}.

When we posted the first version of this paper, we were unaware of previous appearances of this identity in the literature; a related identity had been used by Erdos-Schlein-Yau and by myself and Van Vu for applications to random matrix theory, but to our knowledge this specific identity appeared to be new. Even two months after our preprint first appeared on the arXiv in August, we had only learned of one other place in the literature where the identity showed up (by Forrester and Zhang, who also cite an earlier paper of Baryshnikov).

Peter Denton, Stephen Parke, Xining Zhang, and I have just uploaded to the arXiv a completely rewritten version of our previous paper, now titled “Eigenvectors from Eigenvalues: a survey of a basic identity in linear algebra“. This paper is now a survey of the various literature surrounding the following basic identity in linear algebra, which we propose to call the eigenvector-eigenvalue identity:

Eigenvectors from Eigenvalues: a survey of a basic identity in linear algebra

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