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?

  • $\begingroup$ Please let me know if you agree with my edits. $\endgroup$ – Rodrigo de Azevedo Jun 30 at 19:20
  • $\begingroup$ 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? $\endgroup$ – Blazej Jun 30 at 19:21
  • $\begingroup$ Possibly related: math.stackexchange.com/q/1081446/339790 $\endgroup$ – Rodrigo de Azevedo Jun 30 at 19:30

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