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It's not an animation, but when I saw the visual illustration of eigenvalues/eigenvectors on this page I was blown away: http://en.wikipedia.org/wiki/Eigenvalu

It's not an animation, but when I saw the visual illustration of eigenvalues/eigenvectors on this page I was blown away: http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors I studied linear algebra in college and computed plenty of eigenvalues/eigenvectors, but until I saw that graphic I had no idea what they actually were. I can't believe in retrospect that a textbook would explain them without an illustration like this.

elteto 13y ago
I was also blown away the first time I saw that animation, it really helped to clarify the notion of what eigenvalues and eigenvectors are. However, that is only a geometrical interpretation which is a very particular case, as eigenvalues/vectors appear in many many other applications which don't really have a geometrical representation or meaning. That's probably why books tend to stay away sometimes from particular examples, even though I have to say that this one specifically is very useful.
- haberman 13y ago
  Isn't the geometric formulation equivalent to the algebraic one? What makes the geometric representation any less general? Of course in higher dimensions you can't formulate it in terms of a drawing, but for 2 it seems to capture the entirety of the concept.
  - elteto 13y ago
    Degrees of freedom don't necessarily have to have a geometric meaning in some (many) systems! For example, in basic analysis of mechanical vibrations you can use modal analysis to decompose a coupled system (two harmonic oscillators with viscous dampers in series for example) into two independent systems. The eigenvalues here correspond to the natural frequencies and the eigenvectors are the modal shapes, which give you an idea of how the oscillators actually end up moving after some disturbance. There is some geometrical significance here but is not quite as easy to interpret.
- plus9z 13y ago
  But how do you teach anything without examples? Pedagogically, this stuff is invaluable.
chestervonwinch 13y ago
I made a little interactive web app for visualizing 2-d linear transformations some time ago. My original intention was to see what a 2-d DFT did. Turns out its pretty nifty for seeing the action of eigenvectors, singular matrices, rotations, reflections... anyway, feel free to check it out: http://htmlpreview.github.io/?https://raw.github.com/chestervonwinchester/linear-transforms/master/linear-transforms.html http://htmlpreview.github.io/?https://raw.github.com/chester...

elteto 13y ago

I was also blown away the first time I saw that animation, it really helped to clarify the notion of what eigenvalues and eigenvectors are. However, that is only a geometrical interpretation which is a very particular case, as eigenvalues/vectors appear in many many other applications which don't really have a geometrical representation or meaning. That's probably why books tend to stay away sometimes from particular examples, even though I have to say that this one specifically is very useful.

haberman 13y ago
Isn't the geometric formulation equivalent to the algebraic one? What makes the geometric representation any less general? Of course in higher dimensions you can't formulate it in terms of a drawing, but for 2 it seems to capture the entirety of the concept.
- elteto 13y ago
  Degrees of freedom don't necessarily have to have a geometric meaning in some (many) systems! For example, in basic analysis of mechanical vibrations you can use modal analysis to decompose a coupled system (two harmonic oscillators with viscous dampers in series for example) into two independent systems. The eigenvalues here correspond to the natural frequencies and the eigenvectors are the modal shapes, which give you an idea of how the oscillators actually end up moving after some disturbance. There is some geometrical significance here but is not quite as easy to interpret.
plus9z 13y ago
But how do you teach anything without examples? Pedagogically, this stuff is invaluable.

chestervonwinch 13y ago

I made a little interactive web app for visualizing 2-d linear transformations some time ago. My original intention was to see what a 2-d DFT did. Turns out its pretty nifty for seeing the action of eigenvectors, singular matrices, rotations, reflections... anyway, feel free to check it out: http://htmlpreview.github.io/?https://raw.github.com/chestervonwinchester/linear-transforms/master/linear-transforms.html http://htmlpreview.github.io/?https://raw.github.com/chester...