  The basic result of this note is a statement about the existence of families
of partitions of the set of natural numbers with some favourable properties,
the n-optimal matrices of partitions. We use this to improve a decomposition
result for strongly homogeneous Souslin trees. The latter is in turn applied to
separate strong notions of rigidity of Souslin trees, thereby answering a
considerable portion of a question of Fuchs and Hamkins.
