[haskellandchill]

The visual explanation movement falls flat for me. It's like trying to understand Monads through blog posts. It's great if you already understand the concept to develop your intuition, or if you've never heard of the concept to pique your interest, but it won't help in the intermediate area where you know what you want to know but don't understand it fully. I need to build proofs through incremental exercises to grasp these concepts.

[whatshisface]

As someone who understands eigenfunctions already, I don't understand the pictures either. Here is the best way to think about it: a matrix is a transformation, a composition of rotation, scaling, etc. Eigensets are lines going through the origin that the matrix moves points along. So a rotation would have no eigenvectors because none of the points move in a straight line, while a scaling along the x axis would have an eigenset that was also along the x axis, consisting of the points that were moved straight up or down.

To imagine finding the eigenset, just ask, could I draw a line through 0,0 such that any point I put on it would stay on it after the matrix acted?

[msla]

> So a rotation would have no eigenvectors because none of the points move in a straight line, while a scaling along the x axis would have an eigenset that was also along the x axis, consisting of the points that were moved straight up or down.

Maybe I'm being pedantic, but rotation matrices have eigenvectors and eigenvalues, but the eigenvalues are imaginary because imaginary numbers are rotations in the complex plane.

It's exactly like saying x^2 = -1 has no solutions: It has two solutions, like any other quadratic, but neither of them are real.

In three dimensions, rotation matrices have three eigenvalues, one of them being 1, and the eigenvector corresponding to that eigenvalue is, naturally, the rotation axis.

https://en.wikipedia.org/wiki/Rotation_matrix

[electricslpnsld]

> So a rotation would have no eigenvectors

Rotations have eigenvectors: a 2D rotation has two complex eigenvectors, a 3D rotation has one real and two complex eigenvectors, ...

[whatshisface]

That's a fair and true catch, but I can cover myself by pointing out that the article was only talking about matrices in R^(m x n). ;)

[sakuronto]

When your field of interest is the reals, those complex eigenvectors don't matter.

[haskellandchill]

I want to know it. I have to manipulate the mathematical objects computationally using proof techniques to know it. That just takes time. Thanks though.

[pizza]

Does this gif help? https://commons.wikimedia.org/wiki/File:Eigenvectors.gif

[haskellandchill]

No gif will help. There is no visual explanation that will help was my point.

[Retra]

Surely visual information helps with geometric problems? Our geometric intuition basically evolved to predict the relevance of light patterns on our retinas, and geometry is a language designed to encode these intuitions, so there's no reason to think visual tools presenting geometric facts would inherently fall short.

[haskellandchill]

And yet I still think that. What's up? I'm broken. Good to know.