...is pretty famous by now and helps a lot of people, but there's actually more information in it than the description below it lets on.
It's a two-dimensional plot, with two-dimensional vectors, and the number of eigenvectors is two. Well, the number of families of eigenvectors; the number of colours. By noticing that they're orthogonal to each other, you can imagine that if we were in three dimensions, there'd be a third one, orthogonal to the other two.
The number of eigenvalues is pretty obvious from the maths (the characteristic eqn will always have a lambda^n in it), but that there's the reason graphically. Most people only think of that picture as the behaviour of eigenvectors, but it also allows you to see the space they occupy and their relation to each other.
(oh, and dim(graph) == dim(matrix) because dim(graph)==dim(vectors)==dim(matrix))
While I'm on the subject, I think trying to visualise everything in two dimensions can set you up to make certain mistakes. Most people think that if lines aren't parallel, then they touch once and then diverge. This is true of planes, not lines, in general.
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This picture...
http://en.wikipedia.org/wiki/File:Eigenvectors.gif
...is pretty famous by now and helps a lot of people, but there's actually more information in it than the description below it lets on.
It's a two-dimensional plot, with two-dimensional vectors, and the number of eigenvectors is two. Well, the number of families of eigenvectors; the number of colours. By noticing that they're orthogonal to each other, you can imagine that if we were in three dimensions, there'd be a third one, orthogonal to the other two.
The number of eigenvalues is pretty obvious from the maths (the characteristic eqn will always have a lambda^n in it), but that there's the reason graphically. Most people only think of that picture as the behaviour of eigenvectors, but it also allows you to see the space they occupy and their relation to each other.
(oh, and dim(graph) == dim(matrix) because dim(graph)==dim(vectors)==dim(matrix))
While I'm on the subject, I think trying to visualise everything in two dimensions can set you up to make certain mistakes. Most people think that if lines aren't parallel, then they touch once and then diverge. This is true of planes, not lines, in general.
by which I meant: parallel and orthogonal to the linear transformation, not to each other. same thing for the third one.