So I'm doing a freelance writing job, writing a script for a YouTube video about eigenvectors/values. It took me a while to decide what the focus was going to be, but I finally settled on focusing on how all linear transformations can be thought of as a composition of rotation, scaling, and flipping/reflecting transformations and how eigenvectors are vital to be able to actually do such a decomposition (I'm thinking specifically of eigendecomposition for diagonalizable matrices and polar decomposition and SVD for general matrices, all of which rely on the existence if eigenvectors. However, I'm not planning on getting into the mechanics of how to actually do these decompositions and even necessarily distinguishing between them, since the video is aimed at a general audience. I will be explaining the basics of what a linear transformation is and what a vector is (probably just sticking to the "arrow on the euclidean plane" explanation though). My problem is that I'm stuck on the introduction. I have most of the body written (though it needs revision), but I'm unsure how to introduce the topic to a general audience in a way that will get people immediately interested. What I currently have for the intro is this:
Hey Crazies! [That's how he always introduces videos on his channel] Have you ever wondered how we keep track of how things move? Like, REALLY thought about it? Well, first we need to choose some point that ISN'T moving. This can be whatever we want -- like we've said previously, motion is relative. We call this point the origin and denote it as (0,0). The usual approach is to then the define three vectors of length one, one on the x axis, one on the y axis, and one on the z axis. Then, if we want to see what happens to other vectors are they're moving, we represent them in terms of these three "basis" vectors. (Animation of a particular example of this) . But sometimes, this isn't very convenient. For example, let's say we have a transformation like this (animation of some linear transformation that rotates the standard basis vectors a lot) that's rotating those vectors all over the place. It's hard to even keep track of where they're going, let alone use them to keep track of where everything else is going! We need a better approach. Let's look at the other vectors. Are there any that AREN'T being rotated so much? Well there has to be at least ONE that's not being rotated at all -- the axis of rotation. Are there any others? If there are, we can use them to much more easily keep track of where all the other vectors are moving. Such vectors are called eigenvectors.
However, the guy I'm writing it for said it's too abstract and I agree. Plus, it doesn't really fit the theme I'm going with, which is eigenvectors as a way of breaking complicated transformations into several simple ones.
I think what I need is a specific example/analogy of how eigenvectors can make things easier, but one that will make sense to a general audience who aren't necessarily familiar with linear algebra. He suggested diagonalization and I do like that idea, as it relates very much to eigendecomposition -- after all, eigendecomposition is really just rewriting the matrix in its eigenbasis but with the change of basis matrix and its inverse still acting on it and the diagonalized form of the matrix is just the matrix written in it's eigenbasis. But how do I introduce those ideas to a general audience as an introduction? I need something that will get people immediately interested so they'll want to keep watching.
Any advice is greatly appreciated! :)