2
$\begingroup$

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?

  • 1
    **If $A$ is positive** then this is true for any non-decreasing function in place of $\phi(t)=t^p$. It's a consequence of the [Schur-Horn theorem](http://en.wikipedia.org/wiki/Schur%E2%80%93Horn_theorem). In general, I think you should interpret $|\lambda_j|$ as singular values of $A$, and try the [min-max principle](http://en.wikipedia.org/wiki/Min-max_theorem).2012-12-29

0 Answers 0