The structure of completely positive matrices according to their CP-rank and CP-plus-rank

We study the topological properties of the cp-rank operator cp(A)cp(A) and the related cp-plus-rank operator cp+(A)cp+(A) (which is introduced in this paper) in the set SnSn of symmetric n×nn×n-matrices. For the set of completely positive matrices, CPnCPn, we show that for any fixed p   the set of matrices AA satisfying cp(A)=cp+(A)=pcp(A)=cp+(A)=p is open in Sn∖bd(CPn). We also prove that the set AnAn of matrices with cp(A)=cp+(A)cp(A)=cp+(A) is dense in SnSn. By applying the theory of semi-algebraic sets we are able to show that membership in AnAn is even a generic property. We furthermore answer several questions on the existence of matrices satisfying cp(A)=cp+(A)cp(A)=cp+(A) or cp(A)≠cp+(A)cp(A)≠cp+(A), and establish genericity of having infinitely many minimal cp-decompositions.