For $A \in \mathbb{R}^{n\times n}$, I am wondering if the following inequality holds, $$\text{tr}(A)^2 - \text{tr}(A^2) \leq ||A||_*^2 - ||A||_F^2$$ This is equivalent to the following inequality on the elementary symmetric polynomial, $$\sum_{i<j}\lambda_i\lambda_j \leq \sum_{i<j} \sigma_i \sigma_j$$ where $\lambda_i$ and $\sigma_i$ denote the eigenvalues and singular values of $A$.

This question arises originally from a discussion here. I am still looking for a proof or counterexample to this inequality. Thank you very much for your time.

For $A \in \mathbb{R}^{n\times n}$, I am wondering if the following inequality holds, $$\text{tr}(A)^2 - \text{tr}(A^2) \leq ||A||_*^2 - ||A||_F^2$$ This is equivalent to the following inequality on the elementary symmetric polynomial, $$\sum_{i<j}\lambda_i\lambda_j \leq \sum_{i<j} \sigma_i \sigma_j$$ where $\lambda_i$ and $\sigma_i$ denote the eigenvalues and singular values of $A$.

This question arises originally from a discussion here. I am still looking for a proof or counterexample to this inequality.