"Eigenvectors from Eigenvalues" by Terence Tao et al, and beyond
A recent post "Eigenvectors from Eigenvalues" by Terence Tao et al, https://terrytao.wordpress.com/2019/08/13/eigenvectors-from-eigenvalues/ is viral on social media, starting with https://www.reddit.com/r/math/comments/cq3en0/part_iii_a_physicist_completes_a_linear_algebra/, and popular media, e.g., https://www.quantamagazine.org/neutrinos-lead-to-unexpected-discovery-in-basic-math-20191113, probably due to a nice story associated with the research.
This type of results appears to be known and published many decades ago. As far as I can see from quickly reading the new arxiv report, I reinvented similar result myself in my PhD thesis, an extended version published as a book in 1986 , see p. 64, formula (2.2.26) https://math.ucdenver.edu/~aknyazev/research/papers/old/k.pdf (sorry it is in Russian...) I have found a much earlier appearance, see (2.29) in a beautiful paper https://pdfs.semanticscholar.org/dd58/71b7a8e6c07071bc04e483c09841919f80e3.pdf by Hans Weinberger who cites the even earlier work K. Loewner, :Math. Z . 38,180-181 (1934). (I have had a privilege once communicating with Weinberger in person about this - he was a PhD advisor of my co-author John Osborn.) Terence Tao has replied to my email, that they would include the references in the next revision of their report.
It is an elegant non-evident result. It is reasonably technical, so I am quite puzzled by its popularity on social media. But of course I am very glad to see general public interest in eigenvalues and eigenvectors, which is my core area of expertise. Explaining once to a non-technical person, why my research on eigenvectors in PCA has obtained NSF funding for fighting terrorism, I have come up with the following analogy: "Eigenvectors are vectors, like arrows, so can be used for target shooting bad guys."
P.S. For the pros: The setup is really an operator, not a matrix, eigenvalue problem. The analog of a matrix minor in this operator case is a Ritz projection of the operator to a subspace of co-dimension 1. The coordinates of the vector perpendicular to the subspace in the basis of the eigenvectors can be expressed in terms of the eigenvalues of the original operator and the eigenvalues of its Ritz projection. Everything else can be derived from these expressions. Only works for Hermitian/symmetric operators in Hilbert/Euclidean spaces.
Math+HPC+DS+ML R&D
4 年The main result of ?Terence Tao?et al, paper has a simple and intuitively evident geometric interpretation:? Crossing a 3D ellipsoid centered at the origin and aligned with the coordinate axes by a 2D plane crossing the origin results in a 2D ellipse. The position of the 2D plane relative to the 3D ellipsoid is then uniquely (up to symmetries) determined by 5 numbers: the 2 lengths of the?semi-axes of the?2D ellipse and the 3 lengths of the?semi-axes of the?3D ellipsoid, as in https://commons.wikimedia.org/wiki/File:Indexellipsoid.png? Having the?semi-axes of the 3D ellipsoid?aligned with the coordinate axes, one can derive a formula for the (absolute values of)?coordinates of the unit vector normal to the?2D plane, which is the main result of the paper, implying everything else, as far as I can see.? These formulas can evidently be found in old textbooks on geometry. I have seen one in a 100 y.o. book scanned by Google books.? ? My papers go one step further and derive (from the coordinates of the normal vector) formulas for the (absolute values of) coordinates of the semi-axes of the?2D ellipse relative to the coordinates of the semi-axes of the 3D ellipsoid. The semi-axes of the?2D ellipse are actually Ritz vectors?in the Rayleigh-Ritz method where the trial subspace is a hyper-plane (i.e. co-dimension 1)
Associate Professor at Washington University
5 年Thanks for putting up the links and writing this up. I also shared it on my Twitter feed.