  This paper studies the problem of decomposing a low-rank
positive-semidefinite matrix into symmetric factors with binary entries, either
$\{\pm 1\}$ or $\{0,1\}$. This research answers fundamental questions about the
existence and uniqueness of these decompositions. It also leads to tractable
factorization algorithms that succeed under a mild deterministic condition. A
companion paper addresses the related problem of decomposing a low-rank
rectangular matrix into a binary factor and an unconstrained factor.
