Let A be an \(m\times n\) matrix with nonnegative real entries. The psd rank of A is the smallest k for which there exist two families \((P_1,\ldots ,P_m)\) and \((Q_1,\ldots ,Q_n)\) of positive semidefinite Hermitian \(k\times k\) matrices such that \(A(i|j)={\text {tr}}(P_i Q_j)\) 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 \(\exists {\mathbb {R}}\)-complete. In addition, we give a polynomial time recognition algorithm for matrices of bounded cpsd rank.

令A为具有非负实数项的\(m\times n\)矩阵。 A的psd 秩是存在正半定 Hermitian \(k\times k两个族\((P_1,\ldots ,P_m)\)和\((Q_1,\ldots ,Q_n)\)的最小k \)矩阵使得所有i , j满足\(A(i|j)={\text {tr}}(P_i Q_j)\)。最近的文献中提出了有关相关矩阵不变量的算法复杂性的几个问题：（i）Stark（对于上面定义的 psd 秩），（ii）Goucha，Gouveia（对于无相秩，如果矩阵\（在上述定义中， P_i\)和\(Q_j\)必须为秩一），(iii) 由 Gribling、de Laat、Laurent 提出（对于cpsd 秩，对应于A对称且\(P_i =Q_i\)对于所有i )。我们通过证明相应不变量的决策版本是完整的来解决这些问题。此外，我们还给出了有界 cpsd 秩矩阵的多项式时间识别算法。