Reaching all of R^N by rotations from a linear subspace

[uekstrom]

Hi all,

I have a problem related to quantum mechanical description of vibrational motion in molecules. I would like, for efficiency, to integrate over symmetries (global rotations) of the molecule.

I would like to prove (or disprove) that all points in R^N can be reached by rotations from a linear subspace (let's call it L). In my particular case L has dimension N-3, and my rotations are generated by three matrices T_i, of the form

T_i = \mathrm{diag}(L_i,L_i,..),

where L_x,L_y,L_z are the 3x3 generators of SO(3). T_x is then a block-diagonal matrix with copies of L_x on the diagonal.

So what I want to prove (or disprove) is if, for a given x,

x = exp\left( c_x T_x + c_y T_y + c_z T_z \right) y

has a solution for x from R^N and y from the linear subspace L. L can be chosen arbitrarily, but should not depend on x. From numerical experiments is seems like this rotation is always possible, but I would like to have a formal proof. I will add more details if necessary.

 

[fresh_42]

Let $v\in \mathbb{R}^N$ be any given vector. Since $L$ is a linear subspace, there is an element in $w\in L$ with $|v|=|w|$. Then there is always a rotation which transforms $w$ into $v$. It can be solved in the two dimensional subspace spanned by $v,w$ and then extended in any way to $\mathbb{R}^N$.