I am jealous of the engineering students to come after me. Can you imagine when every advanced textbook is so clear and interactive? Eigenvalues, differential equations, relativity...
...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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I am jealous of the engineering students to come after me. Can you imagine when every advanced textbook is so clear and interactive? Eigenvalues, differential equations, relativity...
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.
I'm even jealous of the engineering students that came before me! :P
Visualization is such a powerful tool. I didn't really understand waves until I saw "Similarities of Wave Behaviour" https://www.youtube.com/watch?v=DovunOxlY1k
watch for one on eigenvalues and vectors soon
Yes!! I cannot wait for the eigenvalues one.
I will watch for it :-)