Chapter 3.11, Problem 46P

We have seen that an orthogonal matrix with determinant 1 has at least one eigen- value $=1,$ and an orthogonal matrix with determinant $=-1$ has at least one eigen-value $=-1.$ Show that the other two eigenvalues in both cases are $e^{i\theta },e^{-i\theta },$ which, of course, includes the real values 1 (when $\theta =0$), and $-1$ (when $\theta =\pi$). Hint: See Problem 9, and remember that rotations and reflections do not change the length of vectors so eigenvalues must have absolute value $=1.$