Exercise 7. of Terence Tao's blog on random matrix, specifically on eigenvalues of Hermitian matrices http://terrytao.wordpress.com/2010/01/12/254a-notes-3a-eigenvalues-and-sums-of-hermitian-matrices/#more-3341 asks to
Show that the $p$-Schatten norms are indeed a norm on the space of Hermitian matrices for every $1\le p\le\infty$.
As I understand it, $p$-norm on the space of matrices of a matrix $A$ is $(\sum_{i,j}A_{ij}^p)^\frac{1}{p}$. To prove the proposition, I multiply a given Hermitian matrix $A$ with a test Hermitian matrix $B$ and take its trace, $$\text{tr}(BA) = \text{tr}(U^\dagger BU\Lambda) = \text{tr}(C\Lambda)$$, where $A=U\Lambda U^\dagger$ is the diagonalization of $A$ with $\Lambda$ the diagonal eigenvalue matrix, and $C$ is the diagonal part of $U^\dagger BU$. I attempt to make the connection between arbitrary test Hermitian matrix $B$ with $\sum_{i,j}|B_{ij}|^q=1$ and arbitrary test diagonal matrix $C$ with $\sum_i|C_{ii}|^q=1$. But I can not proceed.
Perhaps this is not the right route. Can anyone help?