  In this paper we consider the decomposition of positive semidefinite matrices
as a sum of rank one matrices. We introduce and investigate the properties of
various measures of optimality of such decompositions. For some classes of
positive semidefinite matrices we give explicitly these optimal decompositions.
These classes include diagonally dominant matrices and certain of their
generalizations, $2\times 2$, and a class of $3\times 3$ matrices.
