How can I show that $||A||=\sqrt{max(A^2)_{ii}}$ defines a norm on the space of complex self-adjoint matrices $A$?

Question

Here, $(A^2)_{ii}$ denotes a diagonal entry of $A^2$. Using the fact that $A$ is self-adjoint, I managed to prove positivity and absolute homogeneity. But I can't show triangle inequality. I can't figure out how to get around the fact that the maximums for the diagonal entry in $A^2$, $B^2$ and $(A+B)^2$ may not be attained for the same $i$. How can I prove triangle inequality then?

Answer 1

Note that $A_{kk}^2 = e_k^* A^* A e_k = \|A e_k\|^2$, and so $\|A\|_* = \sqrt{\max_k \|Ae_k\|^2} = \max_k \|A e_k\|$.

It is straightforward to demonstrate that this is a norm, the triangle inequality follows from the triangle inequality for Euclidean norm.