Trace of a matrix and its Schatten 3-norm

Question

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.

Answer 1

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

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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 $$ (\varphi(|\lambda_1|),\ldots,\varphi(|\lambda_n|)\prec_w(\varphi(s_1),\ldots,\varphi(s_n)). $$