Re: Rotation in N-space
- From: Gerard Westendorp <westy31@xxxxxxxxx>
- Date: Sun, 15 May 2005 07:05:37 +0000 (UTC)
frisbieinstein@xxxxxxxxx wrote:
> Consider a rigid N-solid in N-space. What are the possible dimensions
> of the axis of rotation, and is it possible to have two simultaneous
> rotations?
>
> In 3-space, in a proper orthagonal rotation matrix one eigenvalue is 1,
> the other two are complex conjugates of the form e^it and e^-it, and
> the determinant is 1. I don't know how to extend this to higher
> dimensions.
>
>
This is how I understand it:
The matrix that transforms coordinates must be real-valued (because
coordinates are real numbers), and the eigenvalues must have norm
1. The only possibilities are eigenvalue 1, eigenvalue -1, and
complex conjugate pairs {exp(ia), exp(-ia)}. The -1 are
reflections, the 1 leave things the same, and the complex conjugate
pairs are the rotations.
So in 3D, you have room only for 1 complex conjugate pair. The
space spanned by its eigenvectors is 2D, the plane orthogonal
to the rotation axis. So there is a 1D space that is left
unchanged by the rotation.
In 4D, you can have 2 pairs. In that case, there are no
non-zero coordinates that are unchanged by the rotation. So
the "axis" is zero dimenional. This is true for all even-
dimensional spaces.
Gerard
.
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