There's a backstory to this paper from the r/math subreddit [1]. One of the authors initially posted the question there, and stumbling upon a similar question on math overflow emailed Terry Tao. And surprisingly, he responded.
Discovered, sort of. His method was nearly the same as the one brought to Tao, but he didn't make the connection to the full correlation between eigenvalues and eigenvectors when dealing with real numbers. Tao used the method to explicitly prove this correlation when it was discovered by Denton, Parke and Zhang.
[1] https://www.reddit.com/r/math/comments/ci665j/linear_algebra...