Let $A = (a_{ij})$ be an $n\times n$ symmetric matrix. Its Schatten norm is defined by $\|A\|_{S^p}^p = \sum_{j=1}^n |\lambda_j|^p$, where $\lambda_j$ are the eigenvalutes of $A$. I am trying to prove that $\sum_{j=1}^n |a_{jj}|^p \leq \|A\|_{S^p}^p$.
I know we can find an orthogonal matrix $Q$ with $Q^T A Q = diag(\lambda_j)$. So, we need to show that the $L^p$ norm of the diagonal of $Q^T A Q$ is at least the $L^p$ norm of the diagonal of $A$. However, I cannot find a way to do this. Any suggestions?