4
$\begingroup$

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.

$\endgroup$

1 Answer 1

9
$\begingroup$

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

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

$\endgroup$
4
  • 2
    $\begingroup$ $\frac{2}{5}$ replaced with $\frac{1}{2}$ works too. $\endgroup$ Jun 3 at 11:27
  • $\begingroup$ 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. $\endgroup$ Jun 3 at 12:50
  • $\begingroup$ Thank you for your answer. $\endgroup$ Jun 4 at 3:09
  • $\begingroup$ @IosifPinelis, thank you. $\endgroup$ Jun 4 at 12:42

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.