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Bulletin of the American Mathematical Society

Published by the American Mathematical Society, the Bulletin of the American Mathematical Society (BULL) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.47.

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Eigenvectors from eigenvalues: A survey of a basic identity in linear algebra
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by Peter B. Denton, Stephen J. Parke, Terence Tao and Xining Zhang HTML | PDF
Bull. Amer. Math. Soc. 59 (2022), 31-58 Request permission

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 $j$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 $j$th row and column by the formula \begin{equation*} |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 ). \end{equation*} 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.
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Additional Information
  • Peter B. Denton
  • Affiliation: Department of Physics, Brookhaven National Laboratory, Upton, New York 11973
  • MR Author ID: 1181364
  • ORCID: 0000-0002-5209-872X
  • Email: pdenton@bnl.gov
  • Stephen J. Parke
  • Affiliation: Theoretical Physics Department, Fermi National Accelerator Laboratory, Batavia, Illinois 60510
  • MR Author ID: 1181367
  • ORCID: 0000-0003-2028-6782
  • Email: parke@fnal.gov
  • Terence Tao
  • Affiliation: Department of Mathematics, University of California, Los Angeles, Los Angeles California 90095-1555
  • MR Author ID: 361755
  • ORCID: 0000-0002-0140-7641
  • Email: tao@math.ucla.edu
  • Xining Zhang
  • Affiliation: Enrico Fermi Institute & Department of Physics, University of Chicago, Chicago, Illinois 60637
  • MR Author ID: 1315731
  • Email: xining@uchicago.edu
  • Received by editor(s): March 4, 2020
  • Published electronically: February 18, 2021
  • Additional Notes: The first author acknowledges the United States Department of Energy under Grant Contract desc0012704 and the Fermilab Neutrino Physics Center.
    This manuscript has been authored by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics. FERMILAB-PUB-19-377-T
    The third author was supported by a Simons Investigator grant, the James and Carol Collins Chair, the Mathematical Analysis & Application Research Fund Endowment, and by NSF grant DMS-1764034
  • © Copyright 2021 American Mathematical Society
  • Journal: Bull. Amer. Math. Soc. 59 (2022), 31-58
  • MSC (2020): Primary 15A18; Secondary 15A42, 15B57
  • DOI: https://doi.org/10.1090/bull/1722
  • MathSciNet review: 4340826