TITLE: Consequences of eigenvector-eigenvalue formula found by studying neutrinos
QUESTION [29 upvotes]: This article describes the discovery by three physicists, Stephen Parke of Fermi National Accelerator Laboratory, Xining Zhang of the University of Chicago, and Peter Denton of Brookhaven National Laboratory, of a striking relationship between the eigenvectors and eigenvalues of Hermitian matrices, found whilst studying neutrinos.
Their result has been written up in collaboration with Terence Tao here. From what I understand, although very similar results had been observed before, the link to eigenvector computation had not been explicitly made prior to now. For completeness here is the main result, "Lemma 2" from their paper:

Let $A$ be a $n × n$ Hermitian matrix with eigenvalues $\lambda_i(A)$ and normed eigenvectors $v_i$.
  The elements of each eigenvector are denoted $v_{i,j}$. Let $M_j$ be the
  $(n − 1) × (n − 1)$ submatrix of $A$ that results from deleting the $j^{\text{th}}$ column and the $j^{th}$ row, with eigenvalues $\lambda_k(M_j)$.
Lemma 2. The norm squared of the elements of the eigenvectors are related to the
  eigenvalues and the submatrix eigenvalues,
$$|v_{i,j}|^2\prod_{k=1;k\neq i}^n(\lambda_i(A)-\lambda_k(A))=\prod_{k=1}^{n-1}(\lambda_i(A)-\lambda_k(M_j))$$

I was wondering what are the mathematical consequences of this beautiful result?
For example are there any infinite dimensional generalisations? Does it affect matrix algorithms or proofs therein? What about singular values?

REPLY [13 votes]: I want to add to the list of places where this identity has been used previously.
For a recent one, see arXiv:1710.02181 “State transfer in strongly regular graphs with an edge perturbation”. 
Equation (2) in Section 2 of this reads
\[
    \frac{\phi(X\setminus a,t)}{\phi(X,t)} = \sum_r \frac{(E_r)_{a,a}}{t-\theta_r}
\]
If the eigenvalue $\theta_r$ is simple. $E_r=z_rz_r^T$ and we have formula (2.3) in the pdf
given in Carlos's reply.
I have used it in a number of places in work on continuous quantum walks, e.g - Lemma 7.1 and Corollary 7.2 in arXiv:1011.0231 "When can perfect state transfer occur?"
It is used implicitly in my paper with Brendan McKay: "Spectral conditions for the reconstructibility of a graph", JCT B (30), 1981, 285–289.
We missed the connection to neutrinos :-(
There is an extensive treatment in Chapter 4 of my book "Algebraic Combinatorics". Chapman & Hall, New York, 1993.
Related identities appear in C. A. Coulson and H. C. Longuet-Higgins: "The electronic structure of conjugated systems I. General theory", Proc. Roy. Soc. London A191 (1947), 39–60. They prove that
if $G$ is a graph with adjacency matrix $A$ and characteristic polynomial $\phi(G,t)$ and $\phi_{ij}(G)$ is the $ij$-entry of $\phi(G,t)(tI-A)^{-1}$, then
\[
    \phi_{ij}(G,t) = \sum_P \phi(G\setminus P,t)
\]
(where the sum is over all paths $P$ in $G$ from $i$ to $j$, and $G\setminus P$ is $G$ with all vertices in $P$ deleted). Further
\[
    \phi_{ij}(G,t) = \sqrt{\phi(G\setminus i,t)\phi(G\setminus j,t)-\phi(G,t)\phi(G\setminus {i,j},t)}.
\]
The connection with eigenvectors arises because if $A=\sum_r \theta_r E_r$ is the spectral decomposition of $A$, then
\[
    \frac{\phi_{ij}(G,\theta_r)}{\phi'(G,\theta_r)} = (E_r)_{i,j}.
\]
Essentially, the identity relates diagonal entries of spectral idempotents of Hermitian matrices to characteristic polynomials of principal submatrices. The off-diagonal entries are related to Green's functions, and I expect that there are papers in the physics literature
where versions of the identity appear.