In Weighted Nuclear Norm Minimization with Application to Image Denoising, it is stated that nuclear norm of a matrix $\mathbf{X}$, given by
$$\|\mathbf{X}\|_{*}=\sum_{i} \sigma_{i}(\mathbf{X})$$
where $\sigma_{i}(\mathbf{X})$ are the singular values, is convex. In the same paper, the weighted nuclear norm, given by
$$\|\mathbf{X}\|_{*,\mathbf{W}}=\sum_{i} w_{(i,i)}\sigma_{i}(\mathbf{X})$$
is non-convex. I am unable to prove this argument with Jensen's inequality. Kindly, help.