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Represenation of SO(2)

  1. Nov 17, 2009 #1
    The usual representation I see of an element of SO(2) is:

    [tex] \left( \begin{array}{ c c } cos(\theta) & sin(\theta) \\ -sin(\theta) & cos(\theta) \end{array} \right) [/tex]

    and it is easy to show that if you make a passive rotation of a cartesian frame by [tex]\theta[/tex] then this matrix will take the comps of an arbitrary vec to those in the new rotated frame.

    However this matrix:

    [tex] \left( \begin{array}{ c c } sin(\theta) & cos(\theta) \\ -cos(\theta) & sin(\theta) \end{array} \right) [/tex]

    is also a valid representation of SO(2), since it has det=1, and transpose equal to inverse. I have played about with a few drawings but just dont see what this actually represents.
     
  2. jcsd
  3. Nov 17, 2009 #2

    D H

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    In your second matrix, theta is the angle between the original y axis and the transformed x axis, with positive theta denoting a clockwise rotation. Alternatively, it is the angle from the transformed x axis to the original y axis, with positive theta denoting a counterclockwise rotation. Neither interpretation is particularly useful or intuitive, which is why it isn't used.
     
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