Note: Most posts on this blog focus on a programming project. This one is more about what I researched while making a programming project, linked at the end. I'm sorry if you like this kind of content less.

Welcome to #RMTotD, prounounced rm-tot'd.

So what is an eigenvector anyways? Good question. For a given transformation matrix (pretty much any square matrix, though typically we use 2x2, 3x3, and 4x4 only), there are a number of eigenvalues up to but not exceeding the size of the matrix (except for when there are infinite eigenvalues). E.g. a 2x2 matrix can have up to 2 eigenvalues, a 3x3 matrix can have up to 3 eigenvalues, a 4x4 matrix can have up to 4 eigenvalues, etc.

But what is an eigenvector, you say as I still haven't actually told you what an eigenvector is despite the fact that I implied that I would tell you.

Well, one way to understand it is that an eigenvector is a vector whose direction is the same whether or not it has been transformed. That is,

\[ \text{Let } b=Ma; \; \hat{a}=\hat{b} \]

Google says that an eigenvector is "a vector that when operated on by a given operator gives a scalar multiple of itself." That scalar is called its eigenvalue, represented by \(\lambda\):

\[ Mv_{\lambda = k} = kv \]

All vectors which match the above definition are eigenvectors, and the collection of all of those vectors (hint: there's an infinite number of them for each eigenvalue. Just change k!) are called the eigenspace. Wow, sounds cool, huh? An eigenspace is easily represented by \( tv_{\lambda=k} \) (for eigenvalue \(k\)) but there are actually a bunch of different ways to represent an eigenspace that all pretty much mean the same thing.

"But wait!" you say frantically. "This is mathdevelopment.tumblr.com! There would never be a post on mathdevelopment.tumblr.com that didn't involve programming!"

Right you are! I produced a little site to help visualize eigenvectors for one of my former teachers to use, but I figured I would post about it here too. You'll need to sit through some more fun properties of eigenvectors before you get the juicy interactive graphs though. After all, this is mathdevelopment.tumblr.com, not randomprogrammingprojectsthatunlockeddecidedtomake.tumblr.com!

Eigenvectors have a cool property where if you have \(n\) eigenvalues in an \(n\text{x}n\) matrix, any point in the relevant \(n\)-dimensional space can be represented as a linear combination of the eigenvectors, and the transformed point will also be a combination of the initial eigenvectors multiplied by their associated eigenvalues:

\[ \text{For} \;\; v_{\lambda=a}+v_{\lambda=b}+\dotsc=v; \] \[av_{\lambda=a}+bv_{\lambda=b}+\dotsc=Mv\]

Cool.