Type: Article
Publication Date: 2021-02-18
Citations: 54
DOI: https://doi.org/10.1090/bull/1722
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 $j$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 $j$th row and column by the formula \begin{equation*} |v_{i,j}|^2\prod _{k=1;k\neq i}^{n}\left (\lambda _i(A)-\lambda _k(A)\right )=\prod _{k=1}^{n-1}\left (\lambda _i(A)-\lambda _k(M_j)\right ). \end{equation*} We refer to this identity as the