  The numerical range of a matrix is studied geometrically via the cone of
positive semidefinite matrices (or semidefinite cone for short). In particular
it is shown that the feasible set of a two-dimensional linear matrix inequality
(LMI), an affine section of the semidefinite cone, is always dual to the
numerical range of a matrix, which is therefore an affine projection of the
semidefinite cone. Both primal and dual sets can also be viewed as convex hulls
of explicit algebraic plane curve components. Several numerical examples
illustrate this interplay between algebra, geometry and semidefinite
programming duality. Finally, these techniques are used to revisit a theorem in
statistics on the independence of quadratic forms in a normally distributed
vector.
