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  1. #1
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    matrix group 2

    Hi there,

    How do I prove that SU(2) = the unit quaternion.... and show that this can be thought of the sphere S^3?
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  2. #2
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    Quote Originally Posted by Loonywoody View Post
    Hi there,

    How do I prove that SU(2) = the unit quaternion.... and show that this can be thought of the sphere S^3?
    SU(2) consists of 2×2 complex matrices that are unitary and have determinant 1. Suppose that U = \begin{bmatrix}a&b\\c&d\end{bmatrix} is such a matrix, where a,b,c,d are complex numbers. The condition det(U) = 1 tells you that ad–bc=1, and therefore U^{-1} = \begin{bmatrix}d&-b\\-c&a\end{bmatrix}. Also, U^* = \begin{bmatrix}\bar{a}&\bar{c}\\\bar{b}&\bar{d}\en  d{bmatrix}. If U is unitary then these two matrices are the same, so comparing matrix entries you see that c=-\bar{b} and d=\bar{a}. Therefore U = \begin{bmatrix}a&b\\-\bar{b}&\bar{a}\end{bmatrix}, where |a|^2 + |b|^2 = 1 (because of the condition det(U) = 1).

    Now let a = x+iy, b = z+iw, where x,y,z,w are real. Then |x|^2 + |y|^2 + |z|^2 + |w|^2 = 1, which is the equation of the sphere S^3 in R^4. Finally, I'll leave you to check that the map \begin{bmatrix}x+iy&z+iw\\-z+iw&x-iy\end{bmatrix} \mapsto x+iy+jz+kw gives an isomorphism from SU(2) to the group of unit quaternions.
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