Eigen-stuff (vectors, values, decomposition) is one of the cornerstone ideas in linear algebra. It’s the stuff that tends to make STEM students “hit their heads against a wall” in frustration. For me, eigenvectors (and determinants) were some of the most challenging ideas found in this branch of mathematics. And what’s more, it’s fundamental to ideas in statistics, physics, engineering, computer science, and more. In this article, I’m going to describe Eigen-stuff as mind-blowingly simple and hopefully it will boost your confidence when tackling trickier topics based on it.

Halloween is nearly upon us and as such, I’m going to use a far-fetched example to illustrate Eigen Decomposition: Let’s say, you’re trick-or-treating down some street — all the odd numbered houses are on the left and even numbers are on the right — and so you bounce from left to right and back again as you make your way down Candy Lane. Neighbors on the left side of the street only give out KitKats, while neighbors on the right side only give out M&Ms. One “round” will include knocking on one door on the left side and then one door on the right side of the street.

For whatever reason, all the neighbors observe some rules, namely, neighbors on the left will give you M&Ms based on the candy you currently have. Namely:

MM = 1*MM + 2*KK #left
KK = 3*MM + 7*KK #right

These rules are expressed by the “linear transformation” captured by the below matrix:

[[1,3],
 [2,7]]

So the question becomes — _after you travel down the entire street (or for an arbitrarily long stretch) _what will your candy bag look like? And how important is your candy bag’s starting composition? Let’s say you start with one of each candy in your bag:

[1,1]
#KK, MM 

#data-science #machine-learning #mathematics #statistics #editors-pick