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

## TL;DR

This survey explores the eigenvector-eigenvalue identity in linear algebra, detailing its history, proofs, and applications in extracting eigenvector components, highlighting its recent rediscovery despite its fundamental nature.

## Contribution

It compiles and reviews the history, proofs, and generalizations of the eigenvector-eigenvalue identity, emphasizing its overlooked significance in linear algebra.

## Key findings

- The eigenvector-eigenvalue identity relates eigenvector components to eigenvalues of minors.
- The identity has been independently rediscovered multiple times since 1834.
- It can be used to determine relative phases of eigenvector components.

## 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 $j^{\mathrm{th}}$ 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 $$ |v_{i,j}|^2\prod_{k=1;k\neq i}^{n}\left(\lambda_i(A)-\lambda_k(A)\right)=\prod_{k=1}^{n-1}\left(\lambda_i(A)-\lambda_k(M_j)\right)\,.$$ We refer to this identity as the \emph{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.