1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    User Avatar
    Staff Emeritus
    Science Advisor

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook