Eli5: what are eigen values and eigen vectors and how is it useful.

Question

Eli5: what are eigen values and eigen vectors and how is it useful.

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Answer 1

Say you have a shape. It can be a circle, a square, a triangle, whatever.

Draw an arrow on that shape.

Now change the shape in some way. You can rotate it, flip it over, stretch bits, squash bits, whatever. All the things you do to it collectively make up the “transformation” you’ve applied to the shape.

If that arrow (*vector*) is still pointing the same direction* after the transformation, then that arrow is an *eigenvector* of the transformation. Transformations can have multiple eigenvectors. For example, imagine *s t r e t c h i n g* your shape out to the left and right. Well, that doesn’t change what direction a left- or right- arrow is pointing, so those are both eigenvectors.

Now, the arrow might have gotten stretched or squished in the process, but as long as it’s still pointing the same direction*, it’s an eigenvector. The *eigenvalue* is the factor by which the eigenvector got stretched or squished – if it got stretched out and now it’s twice as long as it was before, then that eigenvector has a corresponding eigenvalue of 2.

Eigenvectors are an example of an *invariant* – something that doesn’t change amidst a bunch of other things that do change. These often hint at some sort of fundamental property of the underlying system you can exploit to deduce other information from. It’s a bit like, say, the law of conservation of mass – it’s useful because it lets you *derive other facts*, like being able to balance a chemical equation by ruling out the possibility that atoms can just appear or disappear at random.

Eigenvectors are basically the conservation laws of – not chemistry, but the math field of *linear algebra*. If you have a really complicated math or physics thing that can be turned into a linear algebra problem, then it’s useful just in general to keep in mind the law of conservation of eigenvectors(tm).

(* *Technically*, it’s an eigenvector if it *lies on the same line* as before, not necessarily facing the same direction. It can flip to the exact opposite direction and that still counts as an eigenvector; the corresponding eigenvalue will just be negative in that case. Put another way, a vector is an eigenvector if the transformation just manages to scale it by some constant factor (the eigenvalue).)