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Math Help - Matrix group help

  1. #1
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    Matrix group help

    Hiya,
    I've been puzzling over this for ages and I cant find the appropriate proof anywhere... can anyone help me?

    I need to prove that the unitary 1x1 matrix U(1) is isomorphic to the special orthogonal 2x2 matrix SO(2) and that they can be thought of as a circle.

    thanks alot
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  2. #2
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    A unitary 11 matrix is a complex number z such that \bar{z}z = 1, in other words a complex number of modulus 1. This represents a point on the unit circle in the Argand diagram (that's where the circle comes in), and it is of the form z=e^{i\theta}.

    An element of SO(2) is a 22 real orthogonal matrix with determinant 1. Any such matrix must be of the form \begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{bmatrix}, and it represents a rotation through an angle \theta.

    The isomorphism from U(1) to SO(2) is the map that takes e^{i\theta} to \begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{bmatrix}.
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  3. #3
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    Thankyou,
    But there should be a proof for this. That answers too basic.
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  4. #4
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    Quote Originally Posted by Loonywoody View Post
    Thankyou,
    But there should be a proof for this.
    That's your job, not mine.

    You only have to google "SO(2)" to find most of the proofs that you want here (along with a whole lot of links associated with sulphur dioxide).
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