Rank minimization subject to some constraints can be accomplished in many cases through the nuclear norm. \begin{align} \min_{X}.\,\,& \left\|X\right\|_* \\ \text{s.t. }& X\in\mathcal{C} \end{align} where $\mathcal{C}$ is some convex set and $X$ is a Real matrix. According to Recht et al., this can be accomplished by the Semidefinite Program: \begin{align} \min_{X}.\,\,& \operatorname{trace}\frac{1}{2}\left(W_1+W_2\right) \\ \text{s.t. }& X\in\mathcal{C} \\ & \begin{bmatrix} W_1 & X \\ X^T & W_2 \end{bmatrix} \geq0. \end{align}

My question is there a SDP formulation for **complex-valued matrices**???

Thank you