I am working on showing a matrix completion problem is a SDP. Specifically, we want to show \begin{array} \underset{minimize}_{ X\in \mathbb{R}^{m \times n}} & \sum_{(i,j) \in \Omega} ( X_{ij} - Z_{ij} )^2 + \lambda \| X \|_{tr} \end{array} is a SDP.
I know we can formulate $\|X\|_{tr}$ using its dual hence the above problem could be written as:
\begin{array} \underset{minimize}_{X\in\mathbb{R}^{m \times n},W_1\in\mathbb{S^{m}},W_2\in\mathbb{S^n}} &\sum_{(i,j) \in \Omega} ( X_{ij} - Z_{ij} )^2 + \lambda(\text{tr}(W_1)+\text{tr}(W_2))\\ \text{subject to} & \begin{pmatrix} W_1 & \frac{1}{2} X\\ \frac{1}{2}X^{T} W_2\\ \end{pmatrix}\succeq0 \end{array}
But I do not see how to show this is a SDP, either in usual form or canonical form.
Thanks.