E is for Eigenvectors and Eigenvalues
This post is about fear.
Part 1. Fear.
Those words, 'eigenvector' and 'eigenvalue', sounded scary to me for years. I expected the concepts they reference to be hard. And so they were hard. These words / concepts come up in a course called linear algebra. The rest of the course was easy for me, but I struggled with these 'eigen' ideas. So when I was preparing to teach linear algebra for the first time, I got nervous again. It took a while to embrace the idea. Now that stuff doesn't seem so hard. But I remember that the word threw me off, and I know to take it easy when I teach it.
Square roots throw off my algebra students, partly because there's a weird new symbol involved. And partly because the concept that goes with it doesn't mean much to them. (I should have started more slowly than I did this semester, with finding sides of squares that have various areas. I'm kind of zipping through roots and rational expressions, because I wanted to focus more on our last book chapter, using the quadratic formula and graphing parabolas.) They know the √81 is 9, but if they're trying to simplify √162, they'll correctly write √81*2, and then proceed to change that √81 into √9, and then into a 3. It's all just moves in a game that they don't quite understand. And the game is scary.
I remember being very uncomfortable with Greek letters in my first calculus course. There were way too many of them. How was I supposed to memorize twenty-something new symbols?! So I'd read things by saying squiggle every time I saw a Greek letter. That doesn't work if there's more than one of them in the statement you're trying to read. I still slow way down when I try to read things with unfamiliar Greek letters. (Alpha, beta, gamma, delta, and theta are fine. Phi and mu are ok. I'm too lazy to go find the command to write them properly.)
If we always start with interesting problems, instead of with definitions and notations, will we intrigue people enough that fear will never come up? Or is grading going to always introduce the fear factor? What's a good problem for getting students thinking about square roots? Has fear ever come between you and your interest in math?
Part 2. Eigenstuff.
If you don't even want to think about eigenvectors and eigenvalues, stop here. But I'm going to attempt to write about them in a way that makes them feel less scary.
Eigen means something like 'innate' or 'its own' in German. That doesn't sound so bad. They're about not changing in a certain way, so what's a name I could make up that emphasizes the concept of staying the same? (I want to make up my own name for these as a way to make them my own. I think that will help me like them better.) What about home-vectors and home-values, as in 'staying at home'? It's similar to homo- which means same (homogenous, homophone, and homomorphism) and to homeo- which means similar (homeopathy, homeostasis, homeomorphism*), and that's good. It feels homey to me, and that's good too.
Linear algebra deals with vectors (think arrows) and operations on them. For the vector, let's imagine (1,2,3) in R3, our usual 3-dimensional space. (1,2,3) points 1 unit east, 2 units north, and 3 units up. An important concept in linear algebra is linear transformations, which take a vector or group of vectors, and stretch, rotate, or reflect them.
Start with a particular linear transformation A, which is represented by a matrix. The transformation happens by multiplying the matrix A times the vector (say x), getting Ax. Given A, are there vectors that don't change direction, but just stretch or shrink? If there are, those are called eigenvectors (home-vectors), and each one has an eigenvalue (home-value) associated with it, that describes how its length changed. The eigenvalues/home-values usually get labeled with Greek letters, lambda most commonly. I'll use s for stretch (or scalar, if you like). The definition gets us Ax=sx, which says that when the transformation A works on the eigenvector/home-vector x, you get x multiplied by a number (that's what scalar means), so that it just stretches.
Wikipedia has more, of course. (I liked seeing the applications, though most of it was beyond me. And I liked this: They pointed out that if the dimension is 5 or more, there is no method for finding exact values, and round-off error can make numerical methods problematic "because the roots of a polynomial are an extremely sensitive function of the coefficients".)
What's a good problem for getting students thinking about these ideas, before we ever say the E-word?
*Homomorphisms and homeomorphisms are two different things, both mathematical. See wikipedia.