  The Complex Axis theorem states that any endomorphism of a finite-dimensional
complex vector space affords an eigen-vector (or "invariant axis"). A geometric
proof of this geometric result was given by A. de Medeiros, transforming the
endomorphism into a topological self-map with Lefschetz number not equal to
zero. We give a dual version of this proof, which may be more uniform, and does
not rely on the need to do any calculation of an Euler characteristic or
Lefschetz number. A vector field on Projective space is read off directly from
the coordinates ("entries") of the given endomorphism (complex square matrix).
A bordism is defined between such vector fields by means of Stokes' Theorem
applied to a real manifold-with-boundary. This is the principle behind Hopf's
lemma relating the Gauss map and the index of a vector field. All vector fields
of the de Medeiros type are co-bordant to the Milnor-Hopf vector field. This
latter comes from a non-derogatory, real diagonal endomorphism, so clearly
possesses an eigen-vector. Therefore so has the given arbitrary endomorphism.
The main theorem on complex polynomials naturally follows, using the companion
matrix, secular polynomial reciprocity. The geometric Complex Axis derivation
is meant to avoid determinants or "general position" arguments.

Functional Analysis;Differential Geometry