Utility of standard matrix norm

Question

Just thought to myself that after all, $(M_n(\mathbb{R}),+,.)$ is conceptually the same $\mathbb{R}$-vector space as $\mathbb{R}^{n^2}$, so the norm $\sqrt{tr(AA^*)}$ must be equivalent to all the usual $p$-norms on $\mathbb{R}^{n^2}$. So what makes it better ? Is it used on other spaces whose dimension is a perfect square ?

Answer 1

It is consistent with the inner product $\langle A, B \rangle = \operatorname{tr} (AB^*) = \sum_{i,j} [A]_{ij} \overline{[B]}_{ij}$.