Complex Odd-Dimensional Endomorphism and Topological Degree

TL;DR

This paper explores the relationship between odd-dimensional linear mappings, eigenvalues, and topological properties, providing new elementary proofs and linking matrix theory with vector bundle topology.

Contribution

It introduces a novel elementary proof connecting eigenvalues of odd-dimensional mappings to topological invariants, and relates matrix rank theory to vector bundle equations.

Findings

Odd-dimensional real linear maps have real eigenvalues.

A complex map without eigenvectors induces a vector bundle equation.

Topological degree invariance and vector bundle theory lead to key contradictions.

Abstract

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.