EIGENVECTORS FROM EIGENVALUES: A SURVEY OF A BASIC IDENTITY IN LINEAR ALGEBRA

If A is an n x n Hermitian matrix with eigenvalues lambda(1)(A),..., lambda(n)(A) and i, j = 1,..., n, then the jth component v(i,j) of a unit eigenvector v(i) associated to the eigenvalue lambda(i)(A) is related to the eigenvalues lambda(n-1)(M-j),..., lambda(n-1)(M-j) of the minor M-j of A formed by removing the jth row and column by the formula vertical bar v(i, j)vertical bar|(2) Pi(n)(k=1;k not equal i) (lambda(i)(A) - lambda(k)(A)) = Pi(n-1)(k=1) (lambda(i)(A) - lambda(k)(M-j)). 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.

Automated summary

This article introduces a mathematical identity called the eigenvector-eigenvalue identity, which can be used to calculate the relative phases between the components of an eigenvector. The article also provides several proofs and generalizations of this identity.