So there's one part of the explanation on the page that I don't quite get: she uses the example of a coin being turned 360 degrees along some axis as leaving all possible vectors as eigenvectors:
"If you rotate a coin by 360 degrees you preserve all directions and so each direction is an eigenvector. Because no stretching has occurred, all of these eigenvectors have eigenvalue 1. Rotating the coin by 60 degrees destroys all directions and this transformation has no eigenvectors or eigenvalues at all."
But in the 60-degree case, why isn't the axis of rotation an eigenvector? Its direction isn't "destroyed," is it?...
Re:What are eigen values?
by anonymousDan on Thursday August 13, 2009 @02:16PM
(#761366)
Because the original vector is not pointing in the same
direction as the final vector (it is at an angle of 60 degrees to it). If you do a 360 degree rotation you get back to where you started. So any number of successive
full rotations will have eigenvectors, each with eigenvalue
1. If you do a 180 degree rotation, the resulting vector
will be pointing in the opposite direction, and will have
eigenvalue -1.
Because the original vector is not pointing in the same direction as the final vector (it is at an angle of 60 degrees to it).
Mmm, no, the axis around which I rotated the coin didn't change at all, by definition.
Suppose you have a (finite) cloud of points in three dimensions. For example, a bunch of GPS measurements. Lets just imagine that they are roughly elliptical, stretched out like a football. For simplicity we'll subtract the mean from this, so our co-ordinates come from the middle of this cloud.
I can take this set of (x,y,z) points and and find the eigen (which means "characteristic") vectors and values for this set of points. The first eigenvector will point along the long axis of the "football" (right through the pointy bits) because this is the strongest, in some sense, single direction that is represented. It won't be perfectly aligned with a finite number of points...
The second and third eigenvectors will between them form a plane perpendicular to this. So essentially what you've done is found a co-ordinate system that is aligned with the data (modulo some details, this is essentially what a procedure called Principal Component Analysis (PCA) does.
Now that's the vectors, what about the values? The eigenvalues tell you the relative importance of each direction. If you stretch the "football" out, the first eigenvalue, which was associated with the first eigenvector pointing along the long axis, will be larger relative to the others. If you use a basketball, they'll all be the same magnitude.
Does that help?
The idea generalizes a lot from what I've told you, but there are a ton of places you can look up the details if you'd like.