Further ∃R-Complete Problems with PSD Matrix Factorizations

Let A be an m × n matrix with nonnegative real entries. The psd rank of A is the smallest k for which there exist two families ( P 1 , … , P m ) and ( Q 1 , … , Q n ) of positive semidefinite Hermitian k × k matrices such that A ( i | j ) = tr ( P i Q j ) for all i , j . Several questions on the algorithmic complexity of related matrix invariants were posed in recent literature: (i) by Stark (for the psd rank as defined above), (ii) by Goucha, Gouveia (for phaseless rank , which appears if the matrices P i and Q j are required to be of rank one in the above definition), (iii) by Gribling, de Laat, Laurent (for cpsd rank , which corresponds to the situation when A is symmetric and P i = Q i for all i ). We solve these questions by proving that the decision versions of the corresponding invariants are ∃ R -complete. In addition, we give a polynomial time recognition algorithm for matrices of bounded cpsd rank.