ELI5- Eigenvalues and Eigenvectors?

Question

ELI5- Eigenvalues and Eigenvectors?

In: 197

Answer 1

Suppose I take a rubber sheet and stretch it to twice its length vertically, and squeeze it so it’s half its length horizontally. Are there any shapes I can draw on the sheet that would keep the same shape and orientation when I do this, and just get bigger or smaller?

Yes. Vertical and horizontal lines will keep their shape when stretched. Plus a point at the center, but that’s boring. For other kinds of stretches, the lines that keep their shape may be diagonals, but there’s usually two for a rubber sheet.

The directions of constant-shape stretches are the “eigenvectors”, the amount of stretch in those directions are the “eigenvalues”.

This is useful because once we know how points on these lines behave when stretching, we can figure out what happens to *any other point* pretty easily. In my simple horizontal/vertical example, we just have to figure out how far any point is from the center horizontally and vertically, and double or halve that. For other kinds of stretches, we might have to measure diagonal distances.

But nobody cares about rubber sheets. But in many fields, we often deal with physical and mathematical systems that transform quantities the same way this rubber sheet does. One combination of quantities gets twice as big while another combination gets half as big. Eigenvectors are useful for finding simple solutions to complicated “stretching” problems like this, whether the solutions involve 2 variables (equivalent to stretching a 2-d rubber sheet), 3 (stretching a 3-d rubber block), or millions.

And for extra credit, what about rotations? Suppose I *turn* my rubber sheet instead of stretching it, are there any shapes that keep their shape and orientation? Surprisingly the answer is “yes”, if we’re willing to consider eigenvectors and eigenvalues with imaginary numbers in them! Now that probably sounds like mathematical nonsense, but if you play video games, your computer graphics card is solving this sort of mathematical nonsense a trillion times a second.