Recently there has been much interest in "sparsifying" sums of rank one
matrices: modifying the coefficients such that only a few are nonzero, while
approximately preserving the matrix that results from the sum. Results of this
sort have found applications in many different areas, including sparsifying
graphs. In this paper we consider the more general problem of sparsifying sums
of positive semidefinite matrices. We give several algorithms for solving this
problem and describe several applications of these algorithms.