Given a matrix $A \in \mathbb R^{n\times n}$, its nuclear norm is defined as
$$\|A\|_* = \sum_{i=1}^n\sigma_i(A)$$
where $\sigma_i(A)$ is the $i$-th singular value of $A$. Calculating $\sigma_i(A)$ is very involved, but luckily $\|A\|_*$ has the simpler matrix representation
$$\|A\|_* = \mbox{tr} \left( \sqrt{A^TA} \right)$$
which is easy to be computed.
For the weighted nuclear norm, it is defined as
$$\sum_{i=1}^n w_i \sigma_i(A)$$
where $w_i \geq 0$. Assuming that $A \in \mathbb R^{n \times n}$ is a symmetric matrix, how to get its simpler matrix representation?