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James <james545@xxxxxxxxx> writes:
...
>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.
It can be taken to be 1. In fact, the restriction of A to any
A-invariant line L is either the identity I_L or -I_L. If L and
L' are two A-invariant lines with A|L = -I_L and A|L' = -I_L',
then L+L' is an A-invariant plane P on which A = -I|P, and that
P is a rotation plane (on which the rotation is by \pi radians).
Likewise two distinct lines on each of which the restriction of
A is the identity sum to a rotation plane. After doing as much
of this as you can, you're left with 0, 1, or 2 invariant lines,
and in the last case the sum is a plane on which A restricts to
a reflection, so you can't go on.
>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?
My paper in the Monthly, for one.
>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.
Oops. The distinction is in the invariant line (for n odd) or
the last plane (for n even): if A is the identity on that line,
or a rotation on that plane, then A is in SO(n); otherwise, A
is in O(n) but not in SO(n). --As you say just below [now cut].
Notice too, by the way, since that in any plane every rotation
is the composition of 2 (or 0) reflections in a line, it follows
that every element of O(n) is the composition of at most n
reflections in hyperplanes. This is obscurely related (I think)
to the questions Timothy Murphy has been asking about Coxeter
groups, Weyl groups, and so on.
Lee Rudolph
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