  This paper first reviews how anti-symmetric matrices in two dimensions yield
imaginary eigenvalues and complex eigenvectors. It is shown how this carries on
to rotations by means of the Cayley transformation. Then a real geometric
interpretation is given to the eigenvalues and eigenvectors by means of real
geometric algebra. The eigenvectors are seen to be \textit{two component
eigenspinors} which can be further reduced to underlying vector duplets. The
eigenvalues are interpreted as rotation operators, which rotate the underlying
vector duplets. The second part of this paper extends and generalizes the
treatment to three dimensions. Finally the four-dimensional problem is stated.
