Eigenvectors from eigenvalues: A survey of a basic identity in linear algebra
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}. 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 1934). Further, we provide a number of proofs and generalizations of the identity.
- Research Organization:
- Brookhaven National Lab. (BNL), Upton, NY (United States); Fermi National Accelerator Lab. (FNAL), Batavia, IL (United States)
- Sponsoring Organization:
- USDOE Office of Science (SC), High Energy Physics (HEP); National Science Foundation (NSF)
- Grant/Contract Number:
- AC02-07CH11359; SC0012704; DMS-1764034
- OSTI ID:
- 1561548
- Report Number(s):
- FERMILAB-PUB-19-377-T; arXiv:1908.03795; oai:inspirehep.net:1748780
- Journal Information:
- Bulletin, new series, of the American Mathematical Society, Vol. 59, Issue 1; ISSN 0273-0979
- Publisher:
- American Mathematical SocietyCopyright Statement
- Country of Publication:
- United States
- Language:
- English
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