I stumbled across this today:
https://github.com/leopd/geometric-intuition accompanied by an
assertion that you can find the eigenvectors for a hermitian matrix
from its eigenvalues and the eigenvalues of its submatrices.  I have
not yet worked through the details of that, but it sounds plausible
and might be of interest to some of you, here.

But the repository itself covers a lot more ground  than that.

Anyways, the code is python, but a lot of it is fairly straightforward
to re-implement, and there's good english descriptions and
illustrations, also. So this looks like fun.

FYI,
-- 
Raul
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