3 ms·
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 a
by CamperBob 17y ago
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?...
- anonymousDan 17y agoBecause 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.
- CamperBob 17y agoBecause 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.
- jibiki 17y agoYou are doing a rotation in 3 dimensions. Surely the original example meant two dimensions. In 3 dimensions, the axis of rotation is indeed an eigenvector.