Define $\mathcal{M}_n$ and $\mathcal{S}_n$ as the space of $n\times n$ real matrices and $n\times n$ symmetric real matrices, respectively. I want to solve the problem $$ \min_{A\in S_n}\frac{1}{2}\|A-Q\|_F^2+\lambda\|A\|_*, \quad (1) $$ where $Q\in S_n$, $\|\cdot\|_F$ is the Frobenius norm and $\|\cdot\|_*$ the nuclear norm.
In this paper it is proved that $$ D_{\lambda}(Q) = \text{argmin}_{A\in \mathcal{M}_n}\frac{1}{2}\|A-Q\|_F^2+\lambda\|A\|_*, $$ where $$ D_{\lambda}(Q) = U\text{diag}((\sigma_i-\lambda)_{+})V^T,\quad (2) $$ where $Q = U\text{diag}(\sigma_i)V^T$ is the SVD of $Q$.
My problem is that I want to minimize over $\mathcal{S}_n$ and not $\mathcal{M}_n$, and the solution $D_{\lambda}(Q)$ is not symmetric at least $Q$ is semi-positive definite.
Is there a way to use the result (2) in my problem? Or I should use another method to solve (1)?