  In this paper, we introduce a set of block factor-width-two matrices, which
is a generalisation of factor-width-two matrices and is a subset of positive
semidefinite matrices. The set of block factor-width-two matrices is a proper
cone and we compute a closed-form expression for its dual cone. We use these
cones to build hierarchies of inner and outer approximations of the cone of
positive semidefinite matrices. The main feature of these cones is that they
enable a decomposition of a large semidefinite constraint into a number of
smaller semidefinite constraints. As the main application of these classes of
matrices, we envision large-scale semidefinite feasibility optimisation
programs including sum-of-squares (SOS) programs. We present numerical examples
from SOS optimisation showcasing the properties of this decomposition.
