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Lee Rudolph wrote:
I am trying to understand more the groups SO(n) and O(n) geometrically.
I think I have a good understanding of them algebraically.
Well, my first question is : What does the word "rotation" mean in R^n?
Taken from:
http://en.wikipedia.org/wiki/Rotation_group
"By definition, a rotation about the origin is a linear transformation
that preserves the length of vectors, and also preserves the
orientation, or handedness, of space."
...
Then, why does SO(n) give all rotations in R^n?
Be cause, *by definition* the rotations in R^n are the elements of
SO(n).
That isn't really a very satisfactory answer. It certainly raises
the question, *why* is that a good definition? And, in fact, I think
the answer to that question is, it's *NOT* a good definition. Yes,
it characterizes the elements of SO(n), as would a good definition;
but it doesn't relate them in an essential way to the concept of
*rotating*, a process that happens *in time*. That is: up to the
"and also preserves" clause, the definition is defining the "rigid"
part of "rigid motion", but it's ignoring the "motion" part; and
when that clause comes along, it *happens* to catch exactly the
"rigid motions" that can be achieved *within the ambient space*,
but the fact that it does catch them and only them is not immediately
obvious.
Here's another approach, which I greatly favor. Define an orthogonal
transformation of R^n (with its usual inner product) to be a linear
transformation that preserves the length of each vector. If A is
an orthogonal transformation, a 2-dimensional subspace F of R^n is a
"rotation plane" for A if F is A-invariant (i.e., F=A(F)) and the
restriction A|F is a rotation in the usual sense. The following
theorem can be proved in various ways.
THEOREM. Let A be an orthogonal transformation of R^n. If n = 2k+1
then R^n is the orthogonal direct sum of k rotation planes for A and
one A-invariant line L. If n = 2k+2 then R^n is the orthogonal direct
sum of k rotation planes for A and one A-invariant plane P. []
(For a generic A these direct summands are unique.)
Now we can describe SO(n) and its non-trivial coset in O(n) as follows.
The orthogonal transformation A belongs to SO(n) (resp., to O(n)-SO(n))
if and only if the restriction of A to L or P, as the case may be, is
in SO(1) (= {I}) or SO(2), as the case may be. In particular (actually,
equivalently), a special orthogonal transformation is the composition
of k (if n=2k+1) or k+1 (if n=k+2) *ordinary rotations* in pairwise
orthogonal 2-planes (extended by the identity on the orthocomplements);
these ordinary rotations pairwise commute. Even more in particular
(yet, again, equivalently), a special orthogonal transformation *belongs
to a 1-parameter subgroup* of O(n): that is, the standard frame (of
columns of the identity matrix, in order) can be moved *continuously*
and *rigidly* to frame of columns of A (in order). That, I claim,
captures the intended meaning of "rotation" and justifies the Wikipedian
_fiat_.
I got it. Now, here's how I would do it. I would start by defining the
group Isom(n) of all isometries of R^n (it's easy to see that it is a
group) and then I would prove that, given an element _g_ in that group,
_g_ is a linear map iff g(0) = 0. Then I would define O(n) as the
subgroup of Isom(n) whose elements those isometries which are linear.
At this point, I would prove that, given a linear endomorphism _g_ of
R^n, then _g_ belongs to O(n) iff it preserves angles (that is, it
preserves the usual inner product) and that this is also equivalent
to the assertion g^t.g = id. It follows from this fact that, for each
_g_ in O(n), det(g) can only be 1 or -1. At last, I would define SO(n)
as the subgroup of those elements _g_ of SO(N) whose determinant is
equal to 1.
Best regards,
Jose Carlos Santos
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