Really neat paper. Seems pretty unbelievable that something this fundamental has gone unnoticed for so long.
I heard about this a few days ago; it seems like a neat result!
Not clear how useful it is since applying it requires all eigenvalues of all submatrices; there are much easier computational methods to get eigenvectors from eigenvalues, and for theoretical usage it’s not clear when you would know these submatrix spectra either. But I can imagine it being useful in some proof…
(Their “Corollary 3” is a trivial result, though — if an element of an eigenvector vanishes, it is obvious and doubtless well-known that you can delete that row and column from the matrix/vector to obtain an eigenvector of the submatrix with the same eigenvalue.)
It might be interesting to think about the implications of this for Gaussian quadrature schemes, since in that case the quadrature weights are given by the magnitude squared of the first element of each eigenvector of a Hermitian matrix, which is the ideal case for applying their equation (2) — you only need the eigenvalues of a single submatrix. Not as a computational method, but it might say something interesting about the quadrature weights in theory.
For the record, Alan Edelman pointed out that the result has been known for decades, without much hype: https://terrytao.wordpress.com/2019/08/13/eigenvectors-from-eigenvalues/#comment-529031
This reply from Terence Tao is also illuminating:
We are in the process of completely rewriting the paper; at this point it seems more appropriate to present a historical survey of the various places in the literature the identity has appeared (the earliest relevant reference we have currently is a 1934 paper of Loewner, and there are at least 18 other appearances in the literature to our knowledge), and describe the various proofs, generalisations, and applications that we are now aware of. There is also an interesting sociology-of-science aspect to this story which is also worth recording, in particular how it seems that it was not feasible to integrate all the disparate references to this identity in the literature until a popular science article reporting on the identity created enough “common knowledge” to kick off what was effectively a crowdsourced effort to locate all these prior references.
Was gonna say I swear I saw something like this somewhere before :P. Those who don’t read the literature are condemned to rediscover it I guess. Still a nice little note to put out there.
We also had a similar discussion in the Bivector discourse to discuss the geometric algebra proof of it:
It’s not so surprising of a result from a Grassmann.jl perspective, which has the foundation for determinants.