Trace of a matrix and its Schatten 3-norm I'm trying to prove that, given a complex matrix $A$ one always has
$$|\mathrm{tr}(A^3)| \leq \|A\|_3^3,$$
where $\|\cdot\|_3$ denotes the Schatten $3$-norm. However, it seems to me I'm missing a crucial ingredient and that simply unpacking the definitions isn't enough.
 A: There are probably several ways to prove this; I'm going with the first one I found when looking for appropriate inequalities. Let $\lambda_1,\ldots,\lambda_n$ be the eigenvalues of $A$, and $\sigma_1,\ldots,\sigma_n$ the singular values. I'll refer to Theorem II.3.6 in Bhatia's Matrix Analysis, and I'll just state the simplified version we need here: 

Theorem II.3.6 (Weyl's Majorant Theorem) In the above conditions, 
  $$
\sum_{j=1}^n|\lambda_j|^3\leq\sum_{j=1}^n\sigma_j^3. 
$$

Then
$$
|\operatorname{Tr}(A^3)|
=\left|\sum_{j=1}^n\lambda_j^3\right|\leq\sum_{j=1}^n|\lambda_j|^3\leq\sum_{j=1}^n\sigma_j^3=\operatorname{Tr}(|A|^3)=\|A\|_3^3.
$$

In more detail: 


Theorem II.3.6 (Weyl's Majorant Theorem) Let $A\in M_n(\mathbb C)$ with singular values $\sigma_1\geq\cdots\geq\sigma_n$ and eigenvalues $\lambda_1,\ldots,\lambda_n$ arranged so that $|\lambda_1|\geq\cdots|\lambda_n|$. Then for every function $\varphi:\mathbb R_+\to\mathbb R_+$ such that $\varphi(e^t)$ is convex and monotone increasing in $t$, we have 
    $$
(\varphi(|\lambda_1|),\ldots,\varphi(|\lambda_n|)\prec_w(\varphi(s_1),\ldots,\varphi(s_n)). 
$$


