Papers › Eigenvectors from eigenvalues: A survey of a basic identity in linear algebra
Eigenvectors from eigenvalues: A survey of a basic identity in linear algebra
Peter B. Denton, Stephen J. Parke, Terence Tao, Xining Zhang
The archive published only this paper's code-link row. Authors, date and abstract are from arXiv's metadata (CC0), read from the Kaggle arXiv metadata snapshot of 2026-09-12 where its title matched the archive's; the title is the archive's.
If A is an n ×n Hermitian matrix with eigenvalues λ₁(A),…,λₙ(A) and i,j = 1,…,n, then the jᵗʰ component v_(i,j) of a unit eigenvector vᵢ associated to the eigenvalue λᵢ(A) is related to the eigenvalues λ₁(Mⱼ),…,λₙ₋₁(Mⱼ) of the minor Mⱼ of A formed by removing the jᵗʰ row and column by the formula |v_(i,j)|²∏_(k=1;k≠i)ⁿ(λᵢ(A)-λₖ(A))=∏ₖ₌₁ⁿ⁻¹(λᵢ(A)-λₖ(Mⱼ)) . 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.
Code
Repository list and official/mentioned flags are the archive's, frozen 2025-07-28. Reachability, where shown, is from one Syntology probe window (2026-09-16 to 2026-09-18); repositories not probed show nothing. GitHub stars are not tracked.
Code Syntology ran Syntology
Not run by Syntology. Nothing on this page verifies that the listed code works.
Results from the paper archive 2025-07-28
No leaderboard rows for this paper in the archive.
Report a problem or propose a change · a person checks every report against the paper or source before anything changes; decisions are listed on /corrections