  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 $k = 1,2,3,4,5$.
