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 _________________________________________________ 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 ________________________________________________________ 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 ___________________________________________ 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 _____________________________________________ 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