(putting an image of eigen because i love eigenvector very much)

Remember Eigenvector (ei/gen/vec/tors, formatted as they/them/their/theirs)?

I made a design for gen! (this sounds like if ei weren't mine. I made gen.)

Vec design has several references towards eigenvectors (of course).

Ei have an identity matrix, which is analogous to an eigenvector, because applying an identity matrix as a geometric transformation to something doesn't change it at all, much like how an eigenvector is unchanged by a transformation.

Vec design has lots of color patches that resemble hearts as a reference to Undertale souls, which resist during death due to having determination, much like how eigenvectors remain unchanged after a transformation. Ironically, though, the value of an eigenvector's determinant is zero, which sounds like having zero determination, "det = 0" is written on the right side (vec point of view) of vec face.

For example, if one draws lines across vec hair and ears, they form the second part of the equation of the eigenvector.

The arrows on vec huge hair tuft (?) is based on this image.

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.

an eigenvector

useful in linear maps

but kind of boring

  smell of carrion

reeks on my commute to work

wish it’d get cleaned up

  serpents tempt us all

but we know to avoid them

because who likes snakes

  “improper” fractions

accurately show values

so why insult them?

  children are quite dumb

always playing with danger

but they have fun, so

  contaminated

we could have fixed the water

but we all forgot

Hidden markov

Regression trees

Network analyses

Eigenvector and