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    Rotation matrices

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    Consider the matrix

    A = (cos t sin t)
    (-sin t cos t)

    Show that it is unitary
    Show that the eigenvalues are exp(it) and exp(-it)
    find the eigenvectors
    Verify that U'AU is diagonal matrix, where U is the matrix of the eigenvectors.

    Show that since determinant of a matrix is unchanged under unitary change of basis, argue that

    1. det(A) = product of its eigenvalues for any Hermitian or Unitary A
    2. use the invariance of teh trace under teh same transformation and show that Tr(A)= sum of eigenvalues

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    Solution Preview

    see attached files.

    The matrix is unitary since:

    The eigenvevalues are the roots of the characteristic polynomial:

    Which leads to the following eigenvalues:

    The eigenvector associated with

    The last two equations are actually the same, hence we can choose an arbitrary value for x and this will set up the value of y:

    Thus ...

    Solution Summary

    The solution discusses some of the properties of general rotation matrices and show they are unitary matrices