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Math Help - Encoding quaternions as matrices?

  1. #1
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    Encoding quaternions as matrices?

    I've a problem from my Group theory module that's been puzzling me, which is asking me to rotate a vector x (1,0,0) around a vector Ov, where v is (1/sqrt(3), -1/sqrt(3), 1/sqrt(3)), by 60 degrees by representing x as a matrix Z (encoding (0,1,0,0) as a matrix) and computing
    UZU*,
    where U is the unitary matrix encoding the quaternion
    cos30 + vsin30.

    Now, I know how to rotate it using quaternion conjugates, and how to do it using the cross product method, but this terminology is alien to me. What is a Unitary matrix and how is it calculated? Our lecture notes don't cover it, nor do they mention "encoding quaternions as matrices". I have no idea where to start. Any pointers?

    Thanks for any help that is forthcoming.
    Last edited by EuropeanSon; March 20th 2011 at 06:56 AM.
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  2. #2
    A Plied Mathematician
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    I am by no means an expert here, but this link might prove useful. Compare the quaternion method at the end with the matrix multiplication method at the beginning. I can't help you any more than that, though.
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  3. #3
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    Thanks. I got it done in the end, the solution was to represent each dimension of the quaternion as a multiple of specific 2x2 matrices, namely
    1=I
    i=
    [i 0]
    [0 -i]
    j=
    [0 1]
    [-1 0]
    k=
    [0 i]
    [i 0]

    And then add them together to get the appropriate matrix.
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  4. #4
    A Plied Mathematician
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    Ah, the Pauli spin matrices strike again. Glad you got it.
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