Eigenvectors from Eigenvalues – a NumPy implementation

I was intrigued by the recent splashy result showing how eigenvectors can be computed from eigenvalues alone. The finding was covered in Quanta magazine and the original paper is pretty easy to understand, even for a non-mathematician.

Being a non-mathematician myself, I tend to look for insights and understanding via computation, rather than strict proofs. What seems cool about the result to me is that you can compute the directions from simply the stretches (along with the stretches of the sub-matrices). It seems kind of magical (of course, it’s not 😉 ). To get a feel for it, I implemented the key identity in the paper in python and NumPy and confirmed that it gives the right answer for a random (real-valued, symmetric) matrix.

I posted the Jupyter Notebook here.

2 thoughts on “Eigenvectors from Eigenvalues – a NumPy implementation

  1. Thanks for this Corey. I was similarly intrigued by this result and was taking some time to go through it. Although I’ve not had time to go through the proof properly yet (hopefully soon) the application didn’t come to light until I went through your explanation and example. You inspired me to replicate what you did in numpy with R.

    https://scienceofdata.org/2019/11/18/eigenvectors-from-eigenvalues-application/

    It’s not as clean and efficient as your code but I think it gets the job done. Just wanted to thank you for writing this and making the result click in my brain. Keep up the excellent work.

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