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Rotation matrix arbitrary axis?

  1. Feb 28, 2013 #1
    Hello everyone.

    I'm having some trouble with rotation matrices. I'm given three matrices

    [itex]J_1, J_2, J_3[/itex]

    which form a basis for the set of skew-symmetric matrices ([itex]\mathfrak{so}_3[/itex]).
    Further, the matrix exponent function is such that

    [itex]exp[\alpha J_i]=R_i(\alpha)[/itex],

    so taking the exponent of the matrix [itex]J_i[/itex] yields precisely the rotation matrix around the i-th standard basisvector, specified by the angle [itex]\alpha[/itex].

    Now I wish to show that any rotation matrix [itex]R(n,\alpha)[/itex] specified by an axis (unit vector) [itex]n[/itex] and angle [itex]\alpha[/itex], can be written as

    [itex]R(n,\alpha)= exp(\alpha \sum n_i J_i )[/itex].

    Here, [itex]n_i[/itex] are the components of the unit vector n.
    I know I will need the surjectivity of exp for this. Given the surjectivity of exp, how does one show the above is true?
  2. jcsd
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