We all have to look things up once in a while, and I can tell you right now that Wikipedia is the best source for practical information on mathematics. But, there are some others out there that are also worth knowing. Most of these sites are encyclopedic, and not meant for browsing top down, so I’ve used the entry for eigenvector as an example for comparison. (Briefly, an eigenvector of a matrix is a vector that does not change direction if multiplied by that matrix, so \(\bf x\) is an eigenvector of \(\bf A\) if \({\bf Ax}=\lambda\bf x\) for some number \(\lambda\).)
1. Wikipedia (also has a Mathematics portal) is undoubtedly the best source for mathematics on the web. This might very well be a controversial point of view, but it’s true. One of the striking things of the Wikipedia math pages is that they often explain why something is done the way it is. The classical approach taken by mathematicians is to tell you how it is, and let you figure out yourself why it might be useful. Being a mathematician myself, I’ve certainly experienced this a lot during my training. I think that, for example, the linear algebra book by Gilbert Strang also derives its popularity from being different from the usual approach, by the way. The Wikipedia entry for eigenvector is very detailed, with visual illustrations, animations, numerous applications, etc.
2. Wolfram MathWorld is another very good and reliable source, created and maintained by Eric Weisstein. It being maintained by a single author undoubtedly gives it strong coherence, but it contains less information than Wikipedia. It does have the occasional joke, however: what’s the volume of a pizza with thickness \(a\) and radius \(z\)? It’s \(pizza\) (\(\pi z^2a\)) (from Pizza Theorem). The Wolfram MathWorld entry for eigenvector is quite complete, but without illustrations. It does highlight the cool fact that repeatedly applying a matrix to a random vector produces a vector that is proportional to the eigenvector with the largest eigenvalue. Showcasing this interesting property shows that the guy knows his stuff.
3. Mathematics - Stack Exchange is the mathematics site of the popular Stack Exchange group of sites (there are many sites now, the first one was Stack Overflow). Stack Exchange is not really an encyclopedic site, of course, but the question that you’re trying to solve might be on there (or you could ask it yourself), and the best answers are often very insightful. There were 1712 questions containing “eigenvector” at the time of writing.
4. Math Open Reference is a site with a lot of potential. It seems to be, at this time, aimed at high-school mathematics. The strength of this site is interactivity. On most of the pages, there is an interactive demonstration. There is no page on eigenvectors yet, but have a look at the page on the focus points of an ellipse as an example.
5. S.O.S. Math is for you if you seek a hands on approach. Generally speaking, this site is much more practical than the others, with lots and lots of numerical examples that are worked out in detail. The S.O.S. Math entry for eigenvector is actually rather cool, and split into Introduction, Computation of eigenvalues, Computation of eigenvectors, and Complex eigenvectors.
6. Encyclopedia of Mathematics has quite a remarkable history. It originates from a Soviet encyclopedia of mathematics (Математическая энциклопедия in Russian), first published in 1977. From 1987 on, a translated version was published by Kluwer as “Encyclopaedia of Mathematics”. Kluwer also produced a CDROM version, and later put the complete thing online. The Encyclopedia of Mathematics entry for eigenvector is rather brief.
7. PlanetMath was started when Wolfram MathWorld went offline for 12 months following a lawsuit by CRC Press against Wolfram Research in 2000. The PlanetMath entry for eigenvector is really brief, and contains only basic information.
Bonus sites: If you don’t need to look up stuff right now, but have some time to kill, have a look at The Prime Pages, a very cool site with all sorts of information on prime numbers (or what did you expect?). This site exists for as long as I can remember, and I’ve wasted many an hour on it over the years. Another greate site is The On-Line Encyclopedia of Integer Sequences. Have a look at the entry for the primes as an example, or at “Sequence A082912”, which I’ve used in The Harmonic Series. Don’t get fooled by the monospace font, this site contains a wealth of information.
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