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

  1. Feb 19, 2008 #1
    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.
    Last edited: Feb 19, 2008
  2. jcsd
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