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](https://math.stackexchange.com/questions/3117111/bounding-the-coefficients-of-the-characteristic-polynomial-of-a-matrix).
I am still looking for a proof or counterexample to this inequality.



An inequality on elementary symmetric polynomial of eigenvalues
As suggested by Darij in the comments to the OP, the compound matrix $\wedge^2 A$ has eigenvalues $\lambda\_i \lambda\_j$ and singular values $\sigma\_i \sigma\_j$. This result can be found, for example in Theorem 2.16 of 


*Zhan, Xingzhi*, Matrix theory, Graduate Studies in Mathematics 147. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-9491-0/hbk). x, 253 p. (2013). [ZBL1275.15001](https://zbmath.org/?q=an:1275.15001).


[see also](https://www.ams.org/publications/authors/books/postpub/gsm-147). Applying the inequality from [this post](https://math.stackexchange.com/questions/381808/sum-of-eigenvalues-and-singular-values) proves the desired inequality.


