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> Jose_Carlos_Santos?= <jcsantos@xxxxxxxx> writes:
>
> >James 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_.
>
> James, check your gmail account.
>
> Lee Rudolph
>
Dear Lee,
I apologize, but the e-mail address that I have in my profile is incorrect,
mainly because I am wary of posting my e-mail address for everyone to see.
But, assuming that not many will read this specific post, could you please
redirect the e-mail that you sent to : greenthorn@xxxxxxxxx ?
Thank you very much for your response, and it mostly makes sense. I have a few
questions :
1) I read a theorem that was slightly different than the theorem that you
quoted. The theorem I read said that if A is in SO(n) where n = 2k + 1, then
R^n is the orthogonal direct sum of m rotation planes for A and
p A-invariant lines L, where p was not necessarily 1. But you say that p = 1.
I saw the analagous theorem for n = 2k + 2, but I didn't see that there was
only one A-invariant plane. Where can I read the proof of this theorem that
you quote?
2) It's unclear to me how you distinguished SO(n) from O(n) - SO(n). The
second sentence of your last paragraph says "The orthogonal transformation A
belongs to SO(n)
(resp., to O(n)-SO(n))", but then you never write "(resp., O(n) - SO(n))"
again.
Well, I would like to say this : From your theorem, let's say n = 2k + 1. Then
if A is in O(n), then I'd like to say that with respect to some basis of R^n, A
is a block diagonal matrix, where one of the blocks is a 1 x 1 block consisting
of 1 or -1, and the rest of the blocks are 2 x 2 rotation matrix blocks (i.e.
cos(t_i) sin(t_i) -sin(t_i) cos(t_i) in row format). So it seems to me that to
determine whether the matrix is in SO(n) or O(n), it only depends on the 1 or
-1. If the 1 x 1 block is a 1, then A is in SO(n). Otherwise, it's in O(n).
The -1 corresponds to a reflection in one coordinate axis, which is orientation
reversing. The other 2 x 2 blocks correspond to rotations in the respective
planes.
So if n = 2k + 1, then I can say that with respect to some basis, an element A
in O(n) - SO(n) is given by one reflection and 2k rotations in planes. An
element in SO(n) is just given by 2k rotations in planes(since the 1 entry
fixes the last line).
Am I right? I may have a couple more questions as well that I would hope to
ask you.
Thank you so much for your time,
James
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