## Eigenvectors from eigenvalues

… Something whose importance only reblogging can be commensurate What's new

Peter Denton, Stephen Parke, Xining Zhang, and I have just uploaded to the arXiv the short unpublished note “Eigenvectors from eigenvalues“.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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