Subject: Re: two on SO(3)
Date: Wed, 24 Jul 2002 12:40:12 -0400 (EDT)
From: Tom Goodwillie
Not so fast: The sum of those 5 matrices is
0 0 0
0 0 0
0 0 5
In SO(3) the sum of 2 or 3 elements is never zero.
The sum of 4 elements can be zero, for example the identity and
1 0 0 -1 0 0 -1 0 0
0 -1 0 and 0 1 0 and 0 -1 0
0 0 -1 0 0 -1 0 0 1
(This is essentialy the only example.)
I don't know about 5.
Tom Goodwillie
> mean matrix addition, then the result is false. Choose a plane
> and consider the 5 matrices corresponding to rotations in that
> plane by multiples of 2\pi/5. Their sum is zero by symmetry, or
> the origin), the statement is surely wrong. It is not even true in the
> subgroup SO(2) (the circle group), e.g., Uj= rotation by 2j\pi/5
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Subject: Re: two on SO(3)
Date: Wed, 24 Jul 2002 18:47:28 +0200 (CEST)
From: Thomas Schick
The "counterexamples" are not counterexamples: the matrices in
question have a commen eigenvector with eigenvalue one, and their sum is
5 times the projection onto this eigenvector, but not zero.
The point is that the inclusion of SO(2) into SO(3) is a group
homomorphism, but does not respect the ring structure for 2x2 or 3x3
matrices, respectively.
Thomas Schick
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Subject: Re: two on SO(3)
Date: 24 Jul 2002 12:54:16 EDT
From: Jeffrey.A.Strom@dartmouth.edu (Jeffrey A. Strom)
To: dmd1@lehigh.edu
Hi
It seems to me that
1) The statement is clearly wrong in SO(2n) because
if A is in SO(2n) then so is -A.
2) This doesn't kill the possibilities for SO(2n+1).
The counterexample given by Ravenel and Neumann is not
a counterexample: If rotation in the plane is given by
the 2x2 matrix A, then rotation in the plane in R^3
is given by the 3x3 matrix
A 0
0 1,
and the sum of any number of such matrices is nonzero:
just look at the bottom right hand corner.
3) I don't have any ideas on the original question.
Jeff
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Subject: Re: two on SO(3)
Date: Wed, 24 Jul 2002 13:05:23 -0400 (EDT)
From: Glen Takahara
To: Don Davis
Yes, by SO(3) I mean the special orthogonal group in
3 dimensions. I apologize for the misuse of terminology.
Also, addition means matrix addition.
By "rotations in a plane" I take to mean that any
vector on that plane gets rotated to a vector also
on that plane (i.e. the axis of rotation is perpendicular
to that plane). As far as I can tell, the example given
by Dr. Ravenel would produce a sum in which 2 of the
columns would be zero, but the third column could not
be zero. In particular, taking the plane to be the XY-plane
makes the 3rd column in the sum equal to (0,0,5)^t.
It's interesting to me that I've received two responses
so far, both of which state that my assertion is false,
but that I've also heard from a journal editor that a
potential reviewer has said that the result doesn't
surprise him. My impression from this is that the question
is non-intuitive. Despite this, I am seeking intuition on
this matter.
Glen Takahara
Dept. of Math and Stats
Queen's University
takahara@mast.queensu.ca
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Subject: Re: two on SO(3)
Date: Wed, 24 Jul 2002 14:20:46 -0400 (EDT)
From: Tom Goodwillie
On the other hand, I don't believe the statement either.
Here is a counterexample (five elements of SO(3) that add up to zero).
One of them will be the identity I.
The other four, Ai (i=1,2,3,4) will be chosen so that
their sum is a scalar matrix and
they all have trace -3/4,
so that their sum must be -I, the only scalar matrix with trace 4x(-3/4).
Choose vectors a_i, i=1,2,3,4 in R^3 to be the vertices
of a regular tetrahedron centered at the origin.
Let A_i be clockwise rotation around a_i through the angle whose
cosine is -7/8. Then the sum of the A_i will commute with the
symmetries of the tetrahedron, which makes it a scalar matrix,
and the trace of A_i is 1+2x(-7/8) = -3/4.
TG