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James wrote:
Dear all,
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? In R^3,
the map A : R^3 ---> R^3 is a rotation if it preserves distances between points, fixes
the origin, fixes some non-zero vector, and represents a 2-dimensional rotation in the plane
perpendicular to this fixed vector.
So what does "rotation" mean in R^n? Then, why does SO(n) give all rotations
in R^n? I know why SO(3) gives all rotations in R^3 (proof in Artin's algebra page 129).
Also, geometrically, what is O(n)? I would like to learn even more about SO(n) and O(n)
geometrically, but I am not sure which questions to ask. Can you recommend a good
reference for me to learn?
Maybe Gilbert Strang's "Linear Algebra" could be helpful; the
geometric side and the algebraic side are both treated.
If one thinks of O(n) as particular nxn matrices, and
A is in O(n), then one can consider how A transforms
some column vectors u and v. A column-vector is
an nx1 matrix.
u |-> Au ( a column-vector)
v |-> Av
The dot product of u and v is then u^t v or v^t u,
where the superscript t stands for matrix transposition.
To preserve scalar products, which have
a natural geometric interpretation in 2 dimensions,
one requires that the dot product of Au and Av be
the same as that of u and v.
So (Au)^t (Av) = u^t v
So u^t A^t A v = u^t v for all u,v.
If A^t A = I, then u^t A^t A v = u^t I v = u^t v.
This means that if A^t A = I, then scalar products
are preserved. I'm pretty sure that
if scalar products are preserved for all pairs
(u,v), then A^t A = I.
In 4 dimensions, the 4x4 matrix
( cos(a) sin(a) 0 0 )
( -sin(a) cos(a) 0 0 )
( 0 0 cos(b) sin(b) )
( 0 0 -sin(b) cos(b) )
is in SO(4), and I imagine it
as a 2D rotation in one plane,
composed with another rotation,
in general by a different angle, in
a plane perpendicular to the first plane.
David Bernier
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