# An inequality on elementary symmetric polynomial of eigenvalues

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, $$\sum_{i where $$\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.

• Are you assuming $A$ to have positive, or at least real eigenvalues? Commented Feb 20, 2019 at 2:54
• @darijgrinberg No. $A$ is any real valued square matrix. Commented Feb 20, 2019 at 2:55
• Oh, I forgot that the symmetric polynomials are real anyway :) Commented Feb 20, 2019 at 2:55
• I think your inequality is just the inequality from math.stackexchange.com/questions/381808/… , but applied to the compound matrix $\wedge^2 A$ (of size $\dbinom{n}{2} \times \dbinom{n}{2}$) instead of $A$. Can you check? Commented Feb 20, 2019 at 2:58
• What is $\|A\|_*$ ? Commented Feb 20, 2019 at 16:15

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.

see also. Applying the inequality from this post proves the desired inequality.