So I am working on practice problems out of a textbook I am working through over break. One of the problems asks to show that given a complex $n\times n$ matrix $X$ and any orthonormal basis $\{u_1,u_2,...,u_n\}$ of $\mathbb{C}^n$ that the Hilbert-Schmidt norm of the matrix given as $$\| X \|=\bigg(\sum_{j,k=1}^n|X_{jk}|^2\bigg)^{1/2}$$ can be expressed as $$\| X \|^2=\sum_{j,k=1}^n|\langle u_j,Xu_k\rangle|^2$$
I figured that, if we consider the coordinate vectors w.r.t. our basis, we essentially have $$\langle u_j,Xu_k\rangle \rightarrow\langle e_j,Xe_k\rangle$$ where it follows that $$\langle e_j,Xe_k\rangle=X_{jk}$$ However, I am rusty on my linear algebra and I know this is not fully correct. I was thinking that, in order to use the coordinate vectors, I need to change $X$ to be w.r.t. our orthonormal basis but I keep confusing myself in the process.