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Abstract Whether the n × n completely positive cone for n ≥ 5 is facially exposed is an open problem raised by Berman et al. in 2015. In this paper, we give a class of non-exposed faces of the 5 × 5 completely positive cone, which implies that the 5 × 5 completely positive cone is not facially exposed. By embedding the 5 × 5 completely positive cone into the n × n completely positive cone for n > 5 , we further show that the n × n completely positive cone is not facially exposed for n > 5. [ABSTRACT FROM AUTHOR]

It is known that every complex square matrix with nonnegative determinant is the product of positive semi-definite matrices. There are characterizations of matrices that require two or five positive semi-definite matrices in the product. However, the characterizations of matrices that require three or four positive semi-definite matrices in the product are lacking. In this paper, we give a complete characterization of these two types of matrices. With these results, we give an algorithm to determine whether a square matrix can be expressed as the product of k positive semi-definite matrices but not fewer, for k = 1 , 2 , 3 , 4 , 5 . [ABSTRACT FROM AUTHOR]