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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 Summary

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

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 ...

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