Quadratic forms and the expansion and rotations of linear endomorphisms

TL;DR

This paper introduces new quadratic forms for linear endomorphisms in Euclidean space, revealing relationships among eigenvalues and eigendirections through a geometric, rotation-based approach with applications to connection theory.

Contribution

It presents a novel, almost-orthogonal expansion method using two-plane rotations, advancing the geometric understanding of eigenstructure in linear algebra.

Findings

Relations among eigenvalues and eigendirections established

Propositions on complexity and geometric multiplicity provided

Application demonstrated in the context of connection theory

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.