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 馃槈 ). 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.

Simudidactic

auto路di路dact聽n.
A self-taught person.
From Greek聽autodidaktos,聽self-taught聽:聽auto-,聽auto-聽+聽didaktos,聽taught;

+

sim路u路late v.
To create a representation or model of (a physical system or particular situation, for example).
From Latin聽simulre, simult-, from聽similis,聽like;

=
(If you can get past the mixing of Latin and Greek roots)

sim路u路di路dacticadj.
To learn by creating a representation or model of a physical system or particular situation. Particularly, using in silico computation to understand complex systems and phenomena.

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This concept has been floating around in my head for a little while. I’ve written before on how I believe that simulation can be used to improve one’s understanding of just about anything, but have never had a nice shorthand for this process.

Simudidactic inquiry is the process of understanding aspects of the world by abstracting them into a computational model, then conducting experiments in this model world by changing the underlying properties and parameters. In this way, one can ask questions like:

  1. What type of observations might we make if x were true?
  2. If my model of the process is accurate, can I recapture the underlying parameters given the type of observations I can make in the real world? How often will I be wrong?
  3. Will I be able to distinguish between competing models given the observations I can make in the real world?

In addition to being able to ask these types of questions, the simudidact solidifies their understanding of the model by actually building it.

So go on, get simudidactic and learn via simulation!

simudidactic