Sharing the Eigenvectors from Eigenvalues paper with my son

Yesterday I saw a neat tweet from Natalie Wolchover:

Terry Tao & co. have just rewritten their paper about the eigenvector-eigenvalue identity that I covered in Quanta. They now review ~2 dozen independent discoveries of the ID that have come to light since our article took off and analyze the sociology: https://t.co/nYSMzAzvg4 https://t.co/c7bqQY4xLg

— Natalie Wolchover (@nattyover) December 4, 2019

I was excited about the result when I first read Wolcover’s original article, but even more excited about the new paper as, by incredibly lucky coincidence, I’m covering eigenvalues and eigenvectors with my older son right now!

The paper gives a simple example of the “eigenvectors from eigenvalues” formula using this matrix:

Yesterday I had my son compute the eigenvalues and eigenvectors for this matrix, which is a nice exercise for someone who learned about those ideas two days ago! Today we tried to use the formula from the paper.

We began by looking at the formula and discussing the 3×3 matrix:

Next I had him work through the standard calculation for one of the eigenvectors:

Before moving on to the final formula, we needed to get some eigenvalues for one of the special submatrices in the formula. Unfortunately we had a little calculation goof that took a minute to find, but we eventually got the right answers:

Finally, we worked through one example of calculating the value for a component of one of the eigenvectors. This part probably could have been done a bit better by us, but live math isn’t always perfect!

I think this new paper is an incredible lucky break for anyone teaching linear algebra now or in the future. It really isn’t that often that a new math paper has a result that is accessible to young students. It was really fun to share these ideas with my son tonight!