Re: Rotation in N-space
- From: tessel@xxxxxxx
- Date: Tue, 14 Jun 2005 05:22:24 +0000 (UTC)
On Sat, 14 May 2005 frisbieinstein@xxxxxxxxx asked:
> 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
^^^^^^^^^^^^
commuting?
> rotations?
You may ultimately be asking about a higher dimensional analog of the
Euler equations describing a tumbling rigid body in E^3, but right now it
seems you seek the "orthogonal decomposition" of an element of SO(n,R)
into a -commuting- product of "ordinary rotations".
See
Peter M. Neumann, Gabrielle A. Stoy, & Edward C. Thompson,
Groups and Geometry
Oxford University Press, 1994
for a proof of two elementary results which I'll state like this:
Thm. Given any element P of O(n), R^n decomposes (as a euclidean inner
product space) into an orthogonal direct sum of invariant subspaces of P,
each having dimension one or two.
Thm. Wrt a P-adapted basis for R^n, the matrix of P is block diagonal,
with each block either +/-1 (a one by one block) or else the matrix of an
element of O(2) (a two by two block).
IOW, every element of SO(n) has an orthogonal decomposition into a
-commuting- product of ordinary rotations, an even number of which may be
"improper" rotations (to ensure the determinant has value unity).
Note the difference between odd and even dimensions. Elements of SO(2m+1)
typically have m "angles of ordinary rotation", associated with m mutually
orthogonal two dimensional invariant subspaces, plus a one dimensional
fixset, sometimes called the "axis". Elements of SO(2m) typically also
have m angles of ordinary rotation, associated with m mutually orthogonal
invariant subspaces, but with a fixset consisting of the origin only (no
"axis").
> In 3-space, in a proper orthagonal rotation matrix
i.e., for an element of SO(3),
> 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.
For example,
[ cos(a) sin(a) 0 0 0 ]
[ -sin(a) cos(a) 0 0 0 ]
A = [ 0 0 1 0 0 ]
[ 0 0 0 1 0 ]
[ 0 0 0 0 1 ]
has characteristic
(lambda^2 - 2 cos(a) lambda + 1)(lambda^3 - 1)
So we have one triple eigenvalue, unity, and we also have a pair of
complex conjugate simple eigenvalues given by
lambda = cos(a) +/- i sin(a) = exp(+/-ia)
This is an example of what I mean by an "ordinary rotation" in E^5: it has
a fixset of dimension 5-2=3; compare the fixset of dimension 1 for a
generic element. (The fixset of a transformation is the set of all fixed
points.) Note that an ordinary rotation other than the identity always
has one multiplicity n-2 eigenvalue, unity, and one complex conjugate pair
of simple eigenvalues.
Next, consider another "ordinary rotation",
[ 1 0 0 0 0 ]
[ 0 1 0 0 0 ]
B = [ 0 0 cos(b) sin(b) 0 ]
[ 0 0 -sin(b) cos(b) 0 ]
[ 0 0 0 0 1 ]
Now A,B form a commuting pair of ordinary rotations and
[ cos(a) sin(a) 0 0 0 ]
[ -sin(a) cos(a) 0 0 0 ]
AB = BA = [ 0 0 cos(b) sin(b) 0 ]
[ 0 0 -sin(b) cos(b) 0 ]
[ 0 0 0 0 1 ]
which has characteristic
(lambda^2 - 2 cos(a) lambda+1)(lambda^2 - 2 cos(b) lambda) + 1)(lambda-1)
and eigenvalues 1, exp(+/-ia), exp(+/-ib). A generic element of SO(5)
looks like this (if expressed in an "adapted basis"), not like a "ordinary
rotation".
(I factored the characteristic over R; over C it factors further into
linear factors, giving the complex conjugate pairs.)
So for g in SO(5) the possibilities are these:
o g is the identity (five-dimensional fixset),
o g is an ordinary (proper) rotation (three-dimensional fixset),
o g can be orthogonally decomposed into a commuting pair of
ordinary
(proper) rotations (one-dimensional fixset),
o can be orthogonally decomposed into a commuting pair of ordinary
improper "rotations" (one-dimensional fixset).
Here's an example illustrating the last possibility:
[ cos(a) sin(a) 0 0 0 ]
[ sin(a) -cos(a) 0 0 0 ]
[ 0 0 cos(b) sin(b) 0 ]
[ 0 0 sin(b) -cos(b) 0 ]
[ 0 0 0 0 1 ]
Similarly for other dimensions.
HTH,
"T. Essel"
.
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