Eigenvectors from Eigenvalues¶

In the recent paper from Peter B. Denton, Stephen J. Parke, Terence Tao, Xining Zhang, a new proof is presented demonstrating the contruction of the squared norm of the eigenvectors of a Hermitian matrix using only the Eigenvalues and the submatrix Eigenvalues.

From the abstract:

We present a new method of succinctly determining eigenvectors from eigenvalues. Specifically, we relate the norm squared of the elements of eigenvectors to the eigenvalues and the submatrix eigenvalues.

This seems like a cool result. This is an implementation of the key identity in the paper which confirms the result computationally.

https://arxiv.org/pdf/1908.03795.pdf

The key identity they discovered is,