  A linear mapping upon real n-dimensional space, where the dimension n is odd,
has a real eigenvalue-eigenvector pair. The corresponding statement for complex
vector spaces holds true for any dimension n, but should be easy to demonstrate
when n is again odd. Derksen uses this result in an induction chain to prove
the Fundamental Theorem of Algebra using matrix representations. In fact the
question is closely tied to topological issues. We discuss proofs coming from
vector field theory and characteristic classes. But a new proof uses rather
elementary constructions. A complex linear n-mapping without an eigenvector
gives rise to three real 2n-mappings, such that all transformations (mappings)
in their (non-trivial) linear span have full rank (are linear isomorphisms).
The theory of matrices of fixed rank shows that we obtain a 2n-plane bundle
equation over the real projective plane. The equation states that the sum of
trivial line bundles is essentially the same as the (same rank) sum of Hopf
line bundles. To show that this yields a contradiction makes use of the
homotopy theory of vector bundles over a base of low dimension, the
two-dimensional Borsuk-Ulam theorem, and the invariance of topological degree
for self-mappings of an (embedded) two-sphere in three-space.
