A matrix completion problem asks whether the unspecified entries of a partially defined matrix may be chosen so as to achieve a conventional matrix with a desired property. Problems based on a variety of properties have been studied, but the positive definite completion problem has received the most attention, due, in part, to its role in several applications in probability and statistics, image enhancement, systems engineering, geophysics, etc. and to its relation with other completion problems including spectral norm contractions and Euclidean distance matrices (important for the molecular conformation problem in chemistry). Another view of the positive definite completion problem is as a mechanism for addressing a fundamental problem in Euclidean geometry: which potential geometric configurations of vectors (i.e. configurations with angles between some vectors specified) are realizable in a Euclidean space?