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

  • Thread starter uekstrom
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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 [tex]R^N[/tex] 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 [tex]T_i[/tex], of the form

[tex] T_i = \mathrm{diag}(L_i,L_i,..) [/tex],

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

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

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

has a solution for x from [tex]R^N[/tex] 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.
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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##.

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