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

If A is an n \× n Hermitian matrix with eigenvalues\λ₁(A),\,\λₙ(A) and i,j = 1,\,n, then the^ component vi,j of a unit eigenvector vᵢ associated to eigenvalue \λᵢ(A) is related to the eigenvalues\λ₁(Mⱼ),\,\λₙ₋₁(Mⱼ) of the minor Mⱼ of A formed by the j^ row and column by the formula |vi,j|²\∏k=1;k\≠ⁿ\(\λᵢ(A)-\λₖ(A)\)=\∏ₖ₌₁ⁿ⁻¹\(\λᵢ(A)-\λₖ(Mⱼ)\)\\,. refer to this identity as the -eigenvalue identity and how this identity can also be used to extract the relative phases between components of any given eigenvector. Despite the simple nature of this and the extremely mature state of development of linear algebra, this was not widely known until very recently. In this survey we describe many times that this identity, or variants thereof, have been discovered rediscovered in the literature (with the earliest precursor we know of in 1834). We also provide a number of proofs and generalizations of identity.