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

Abstract:

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 eigenvector-eigenvalue identity and show how this identity can also be used to extract the relative phases between the components of any given eigenvector. Despite the simple nature of this identity and the extremely mature state of development of linear algebra, this identity was not widely known until very recently. In this survey we describe the many times that this identity, or variants thereof, have been discovered and rediscovered in the literature (with the earliest precursor we know of appearing in 1834). We also provide a number of proofs and generalizations of the identity.