How to get a simpler matrix representation of weighted nuclear norm?

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?

• Please let me know if you agree with my edits. – Rodrigo de Azevedo Jun 30 at 19:20
• I doubt such formula exists. On the other hand it is possible to get matrix representation for norms of the type $\left( \sum_{i=1}^n \sigma_i(A)^p \right)^{\frac{1}{p}}$. Would that be of interest to you? – Blazej Jun 30 at 19:21
• Possibly related: math.stackexchange.com/q/1081446/339790 – Rodrigo de Azevedo Jun 30 at 19:30