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Comment by darwingr | Hacker News Reader

This really would have been harder for me to understand had I not taken linear and abstract algebra courses a few years ago. That area of maths reused common words like "rotation" but with more generalized definitions, which made it was jarring and confusing to hear and take in at the time. When someone said the word "rotate" my mind as if by reflex was already trying visualize a 3d or 2d rotation even when it made no sense for the problem at hand. Being an English speaker my whole life I thought I understood what a rotation was or could be but I didn't.

Same goes for what's being alleged here: Is there even a way to visualize this that makes mathematical sense? What will be the corollaries to this discovery simply as a result of what the mathematics of rotations will dictate?

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Same goes for the ordinary English word "Eigenvector".

Reminds me of how orthogonally polarized waves can inhabit the same bit of space without interfering with each other (and can be cleanly separated later using 2 polarized filters at 90 degrees).

Eigenvector is actually a Denglish word (half German, half English), eigen is the German part, vector the English part.

"After my one course in linear algebra, I knew eigenvectors and eigenvalues like the back of my head. If your instructor was anything like mine, you recall solving problems involving eigendoohickeys, but you never really understood them."

-- Jonathan Richard Shewchuk, from An Introduction to the Conjugate Gradient Method Without the Agonizing Pain

And yet the main image on the article illustrates a 45 degree rotation along an axis.

From what I understand, you are saying this rotation is non-intuitive. Could you elaborate more or share some relevant links?

I would say that one can still interpret it geometrically, but the difficulty is that it's not in 2D or 3D, but in a higher dimensional space.

Imagine you have a population of 100 neurons. To each one you associate a real number representing its activity. So a particular snapshot of firing would have 100 real values. It's a vector in a 100 dimensional space. Each individual neuron is one dimension. The high dimensionality of the space is what makes it unintuitive. But, after some familiarization you learn heuristics for reasoning about high-dimensional spaces and combine that with your 3D spatial intuition.

The 45 degree rotation is just some nice art, but you could think of it as representing a projection down to 2D.

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