Eigenvectors from eigenvalues: A survey of a basic identity in linear algebra

@article{Denton2019EigenvectorsFE,
  title={Eigenvectors from eigenvalues: A survey of a basic identity in linear algebra},
  author={Peter B. Denton and Stephen J. Parke and Terence Tao and Xining Zhang},
  journal={arXiv: Rings and Algebras},
  year={2019}
}
If $A$ is an $n \times n$ Hermitian matrix with eigenvalues $\lambda_1(A),\dots,\lambda_n(A)$ and $i,j = 1,\dots,n$, then the $j^{\mathrm{th}}$ component $v_{i,j}$ of a unit eigenvector $v_i$ associated to the eigenvalue $\lambda_i(A)$ is related to the eigenvalues $\lambda_1(M_j),\dots,\lambda_{n-1}(M_j)$ of the minor $M_j$ of $A$ formed by removing the $j^{\mathrm{th}}$ row and column by the formula $$ |v_{i,j}|^2\prod_{k=1;k\neq i}^{n}\left(\lambda_i(A)-\lambda_k(A)\right)=\prod_{k=1}^{n-1… Expand

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