Eigenvectors from eigenvalues

… Something whose importance only reblogging can be commensurate

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Peter Denton, Stephen Parke, Xining Zhang, and I have just uploaded to the arXiv the short unpublished note “Eigenvectors from eigenvalues“. This note gives two proofs of a general eigenvector identity observed recently by Denton, Parke and Zhang in the course of some quantum mechanical calculations. The identity is as follows:

Theorem 1 Let $latex {A}&fg=000000$ be an $latex {n times n}&fg=000000$ Hermitian matrix, with eigenvalues $latex {lambda_1(A),dots,lambda_n(A)}&fg=000000$. Let $latex {v_i}&fg=000000$ be a unit eigenvector corresponding to the eigenvalue $latex {lambda_i(A)}&fg=000000$, and let $latex {v_{i,j}}&fg=000000$ be the $latex {j^{th}}&fg=000000$ component of $latex {v_i}&fg=000000$. Then

$latex displaystyle |v_{i,j}|^2 prod_{k=1; k neq i}^n (lambda_i(A) – lambda_k(A)) = prod_{k=1}^{n-1} (lambda_i(A) – lambda_k(M_j))&fg=000000$

where $latex {M_j}&fg=000000$ is the $latex {n-1 times n-1}&fg=000000$ Hermitian matrix formed by deleting the $latex {j^{th}}&fg=000000$ row and column from $latex {A}&fg=000000$.

For instance, if we have

$latex displaystyle A = begin{pmatrix} a & X^*…

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