Quadratic forms and the expansion and rotations of linear endomorphisms

Abstract

New expansionary and rotational quadratic forms are constructed for $E^n$-endomorphisms. Relations amongst the various eigenvalues, eigendirections and matrix invariants are established, including propositions on complexity and geometric multiplicity. The underlying construction involves a novel, almost-orthogonal expansion based on two-plane rotations. The development is strongly geometric in flavour and has application to the theory of connections, of which the Frenet case on $E^3$ is given as a model.