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?
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*Homomorphisms and homeomorphisms are two different things, both mathematical. See wikipedia.
Posts
Sue,
ReplyDeleteMy first thought is making sure students have worked with natural, familiar examples with eigenvectors like rotations around some axis in R^3, scaling, etc. Get them to see which examples are easier to work with and why.
Also: probably you ask them to practice a bunch of matrix multiplication. How about a problem where they are given a concrete matrix A with an eigenvector w = (1,2,3) with eigenvalue 2. (Make up an example.) Then ask them to find AAAAAw. This will be very tedious until/unless they appreciate the neat properties you want them to see.
There could even be a choice of problems, where some are not too bad like that, and some are very tedious.
Sounds like you're doing really well! I hope maybe this helps a little. And I agree that the names are intimidating!
-David
My example above could expand into easier iterating of matrix multiplication even for vectors that aren't eigenvectors, if they're linear combinations of eigenvectors.
ReplyDeleteEigenvectors make computations easier even if you aren't iterating a matrix, but are working with the same matrix for awhile. Writing vectors as linear combinations of eigenvectors leads to switching to a basis of eigenvectors. That's a really important application of eigenvectors, but you can build up to it in pieces, or only in part.
If you want to initially avoid the e-words initially (you can't avoid them forever, since they are standard terminology) you could call them the "characteristic" values and vectors.
ReplyDeleteYou might also want to start what David said about concrete matrices with diagonal matrices, which have obvious eigenvalues/vectors, then show that if you know the eigenvalues/vectors of any non-singular matrix, working with it becomes nearly as easy as working with a diagonal matrix.
Sorry, I should have said: I'm not going to be teaching linear algebra any time soon. I taught it in Michigan, and wasn't very happy with how it went. I haven't yet taught it at my college in California. I will some day, but not yet.
ReplyDeleteI think I wanted the example problem so I could learn more myself. In the community college linear courses, eigenstuff is right at the end, so we don't get into it very deeply. I think I want to learn (or is it relearn?) more of this stuff before I get close to teaching it.
Ahh, David, I just saw something. I had noticed that rotations (in R2) won't have any eigenvectors, because everything changes direction. But a rotation in R3 needs an axis, and that direction will be an eigenvector.
I know, very basic. Like I said, it's been a while. I must have worked with this all in the linear course I took while in grad school (senior level course), but I don't really remember any of it. I just remember that I liked it. (It was 24 years ago...)
I'm 54, and still enjoying learning math. I like taking things slowly, I think. ;^)
Blaise, my change of vocabulary was just for myself, really. When I get around to teaching this course, I imagine I'd start with a problem that makes us think about eigenstuff first (without mentioning it).
ReplyDeleteAnd then I'd mention it, and give both the conventional words and my own. If the students got hooked on my words, I'd use both mine and the proper ones. But I suspect they'd just want the proper ones.
I laughed when I saw your blog name, David. Eigensomething is great. And that's about how much I got out of your post. I can vaguely see that eigenstuff might have something to do with what you're working on. :^)
Thank you both for replying. I haven't been sure if these math alphabet posts have been interesting people. And I have very little idea of who my audience is. I just have fun writing about what intrigues me. I wonder if I could do a poll, and find out a bit about who's reading my blog...
@BP: I'm not sure you really mean "non-singular" (which means invertible, not diagonalizable).
ReplyDelete@Sue: Thanks for taking a look at my "blog" -- I know that stuff doesn't make much sense, but I wanted to put something there. My work more recently does not involve eigenvectors, but data analysis instead--the purpose was really for putting up fun data analysis projects I did outside of work, but I've been a bit too lazy.
Sue: About rotations--yeah, exactly.
ReplyDeleteBut at the 'next level', you can think of your matrix as having a complex eigenvalues and eigenvectors, even though the numbers in the matrix are real. And that can be really useful.
http://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspace#Rotation_matrices_on_complex_vector_spaces
Suppose that x people in your state live in the city, and y people live in the country. Each year, 10% of the people who lived in the country last year will move to the city this year; and each year, 5% of the people who lived in the city last year will move to the country.
ReplyDeleteSo if x_n and y_n are the new city and country populations, and x_o and y_o the old ones, then
x_n = (0.95)x_o + (0.10)y_o
y_n = (0.05)x_o + (0.90)y_o
which you could write as a matrix. Now, start with any city and country population you like, and multiply by the matrix to find the population next year, and the next, and the next... You will find the populations approaching an equilibrium.
When the populations are at equilibrium, multiplying by the matrix to get next year's population will give the same population as this year. That is, the equilibrium populations are an *eigenvector* of the matrix, with eigenvalue 1.
You can solve algebraically for this fixed vector by using the fact that for the equilibrium values x_e and y_e, you have x_e=x_o=x_n, i.e.
x_e = (0.95)x_e + (0.10)y_e
y_e = (0.05)x_e + (0.90)y_e
(You'll end up with two copies of the same equation: this only gives the *ratio* of x_e and y_e. To find the values you need to know the total population you started with.)
You can find more examples like this (and more realistic ones) by looking up "Markov chains".
Google uses a similar sort of reasoning to do its ranking of webpages. If you look up "Google eigenvector" you can find two or three good expository articles on this.
Another completely awesome post! I had a matrices class when I majored in physics and these eigen words scared me, too. Your explanations are much better than any I was ever taught. Well done!
ReplyDelete@Andrew, oh yeah, I remember Markov chains. So anything that can be described that way, if it has an equilibrium, will have an eigenvalue of 1 and an eigenvector representing the equilibrium state. I agree that that's a good way to start.
ReplyDelete@Paula, thanks for letting me know you liked the explanation. I was trying for something that would make sense even for folks who haven't ever looked at any of this.