tag:blogger.com,1999:blog-5303307482158922565.post1018421065830406149..comments2024-03-22T13:39:55.941-07:00Comments on Math Mama Writes...: My Math Alphabet: E is for Eigenvectors and EigenvaluesSue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.comBlogger10125tag:blogger.com,1999:blog-5303307482158922565.post-20945060910909646872010-12-01T17:31:16.484-08:002010-12-01T17:31:16.484-08:00@Andrew, oh yeah, I remember Markov chains. So any...@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.<br /><br />@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.Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-20347242343864359082010-12-01T16:04:44.298-08:002010-12-01T16:04:44.298-08:00Another completely awesome post! I had a matrices...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!Paulanoreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-43188361181147004682010-11-29T20:26:10.697-08:002010-11-29T20:26:10.697-08:00Suppose that x people in your state live in the ci...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. <br /><br />So if x_n and y_n are the new city and country populations, and x_o and y_o the old ones, then<br /><br />x_n = (0.95)x_o + (0.10)y_o<br />y_n = (0.05)x_o + (0.90)y_o<br /><br />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. <br /><br />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.<br /><br />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.<br /><br />x_e = (0.95)x_e + (0.10)y_e<br />y_e = (0.05)x_e + (0.90)y_e<br /><br />(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.)<br /><br />You can find more examples like this (and more realistic ones) by looking up "Markov chains".<br /><br />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.Andrewhttps://www.blogger.com/profile/00605171970083004552noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-18414237834412351592010-11-29T16:41:38.537-08:002010-11-29T16:41:38.537-08:00Sue: About rotations--yeah, exactly.
But at the &...Sue: About rotations--yeah, exactly.<br /><br />But 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.<br /><br />http://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspace#Rotation_matrices_on_complex_vector_spacesDavidhttps://www.blogger.com/profile/10565910956857563935noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-61434635371086921772010-11-29T16:35:43.298-08:002010-11-29T16:35:43.298-08:00@BP: I'm not sure you really mean "non-si...@BP: I'm not sure you really mean "non-singular" (which means invertible, not diagonalizable).<br /><br />@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.Davidhttps://www.blogger.com/profile/10565910956857563935noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-8664390308997107992010-11-28T16:05:09.156-08:002010-11-28T16:05:09.156-08:00Blaise, my change of vocabulary was just for mysel...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). <br /><br />And 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.<br /><br />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. :^)<br /><br />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...Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-81586276530502137762010-11-28T15:49:35.664-08:002010-11-28T15:49:35.664-08:00Sorry, I should have said: I'm not going to be...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.<br /><br />I 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.<br /><br />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. <br /><br />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...)<br /><br />I'm 54, and still enjoying learning math. I like taking things slowly, I think. ;^)Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-64704492026041057172010-11-28T15:26:06.935-08:002010-11-28T15:26:06.935-08:00If you want to initially avoid the e-words initial...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.<br /><br />You 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.Buddha Buckhttps://www.blogger.com/profile/17167036913705912859noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-67307792741120261532010-11-28T14:44:09.167-08:002010-11-28T14:44:09.167-08:00My example above could expand into easier iteratin...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. <br /><br />Eigenvectors 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.Davidhttps://www.blogger.com/profile/10565910956857563935noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-16258311620351675082010-11-28T14:30:12.692-08:002010-11-28T14:30:12.692-08:00Sue,
My first thought is making sure students hav...Sue,<br /><br />My 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. <br /><br />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. <br /><br />There could even be a choice of problems, where some are not too bad like that, and some are very tedious. <br /><br />Sounds like you're doing really well! I hope maybe this helps a little. And I agree that the names are intimidating! <br /><br />-DavidDavidhttps://www.blogger.com/profile/10565910956857563935noreply@blogger.com