# An inequality for certain positive-semidefinite matrices

Suppose that $$G=(G_{ij})$$ is a positive-semidefinite symmetric matrix with the diagonal entries all equal $$1$$ and all off-diagonal entries $$\le0$$. Does it then necessarily follow that $$\sum_{i,j}(G^5)_{ij}\le\sum_{i,j}(G^3)_{ij}\,?$$

This is true if it is additionally assumed that all off-diagonal entries of $$G$$ are the same.

If I have not committed any mistake, please, find below a counter-example.

Counter-example. Let $$G\in \mathbb{S}^3_{+}$$ be defined by $$G = \begin{pmatrix} 1 & -\frac{2}{5} & 0 \\ -\frac{2}{5} & 1 & -\frac{2}{5}\\ 0 & -\frac{2}{5} & 1 \end{pmatrix}.$$

• $\frac{2}{5}$ replaced with $\frac{1}{2}$ works too. Commented Jun 3, 2023 at 11:27
• Mark, thank you for the edit. @PeterMueller, indeed. And it works for 1/3 as well, but it fails (i.e., the inequality holds true), e.g., for $1/5$. Not clear at this point for me what would be a 'good' sufficient condition for this inequality to hold. Commented Jun 3, 2023 at 12:50
• Thank you for your answer. Commented Jun 4, 2023 at 3:09
• @IosifPinelis, thank you. Commented Jun 4, 2023 at 12:42