  Let ${\cal S}_+^n \subset {\cal S}^n$ be the cone of positive semi-definite
matrices as a subset of the vector space of real symmetric $n \times n$
matrices. The intersection of ${\cal S}_+^n$ with a linear subspace of ${\cal
S}^n$ is called a spectrahedral cone. We consider spectrahedral cones $K$ such
that every element of $K$ can be represented as a sum of rank 1 matrices in
$K$. We shall call such spectrahedral cones rank one generated (ROG). We show
that ROG cones which are linearly isomorphic as convex cones are also
isomorphic as linear sections of the positive semi-definite matrix cone, which
is not the case for general spectrahedral cones. We give many examples of ROG
cones and show how to construct new ROG cones from given ones by different
procedures. We provide classifications of some subclasses of ROG cones, in
particular, we classify all ROG cones for matrix sizes not exceeding 4. Further
we prove some results on the structure of ROG cones. We also briefly consider
the case of complex or quaternionic matrices. ROG cones are in close relation
with the exactness of semi-definite relaxations of quadratically constrained
quadratic optimization problems or of relaxations approximating the cone of
nonnegative functions in squared functional systems.
