  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.
