**Problem**. Let be orthogonal and suppose that Find

**Solution**. Since is orthogonal, its eigenvalues have absolute value and it can be be diagonalized. Let be a diagonal matrix such that for some invertible matrix Then

We claim that the eigenvalues of are for some Well, the characteristic polynomial of has degree three and so it has either three real roots or only one real root. Also, the complex conjugate of a root of a polynomial with real coefficients is also a root. So, since the eigenvalues of are either all which is the case or two of them are and one is which is the case or one is and the other two are in the form for some So

Note that given the matrix

is orthogonal, and