A real explanation for imaginary eigenvalues

Abstract 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 the necessary tools from real geometric algebra are introduced and a real geometric interpretation is given to the eigenvalues and eigenvectors. The latter are seen to be two component eigenspinors which can be further reduced to underlying vector duplets. The eigenvalues are interpreted as rotors, which rotate the underlying vector duplets. The second part of this paper extends and generalizes the treatment to three dimensions. The final part shows how all entities and relations can be obtained in a constructive way, purely assuming the geometric algebras of 2-space and 3-space.