Faces of the 5 x 5 completely positive cone

The full characterization of extreme rays of the 5 x 5 copositive cone provides an important means to study faces of the 5 x 5 completely positive cone. The maximal faces of the 5 x 5 completely positive cone determined by the Horn matrix or by a Hildebrand matrix can be easily described. In this paper, we study the subfaces of these maximal faces. Weprove that most of the subfaces on these maximal faces are exposed. However, a class of subfaces, which are on the maximal face determined by the Horn matrix, are not exposed. Therefore, besides the class of non-exposed faces appearing in a recent paper by the author, another family of non-exposed faces are presented in this paper to show again that the n x n completely positive cone is not facially exposed for n = 5. In this paper, we also prove a necessary and sufficient condition for a strictly positive matrix, which is on the maximal face determined by the Horn matrix, to have a unique cp factorization.