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Suppose I have a matrix $A,$ and I symmetrize it by the following construction:

$$ B = A + A^t - A \circ A^t,$$ where $\circ$ is the Hadamard (elementwise) product. Are there any reasonable hypotheses on $A$ which make $B$ positive semi-definite?

In the case where $A$ is diagonal, the condition is that the entries are between $0$ and $2.$

Igor Rivin
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    Something like Zhan, Xingzhi. "Inequalities for the singular values of Hadamard products." SIAM Journal on Matrix Analysis and Applications 18.4 (1997): 1093-1095. might be related? – Vezen BU Mar 01 '23 at 08:45
  • @VezenBU Thank you, I will check it out! – Igor Rivin Mar 01 '23 at 11:40
  • You are welcome. I just think that we can approach this problem w.r.t eigenvalues and some spectral results on the Hardmard product might be useful. But I am actually not sure... since I also know that we cannot really have a closed-form on the eigenvalues. There are also several related problems on this site. See https://math.stackexchange.com/search?q=Hadamard+eigenvalue, e.g., https://math.stackexchange.com/questions/711918, https://math.stackexchange.com/questions/4087520. – Vezen BU Mar 01 '23 at 12:56
  • See also the comments in https://math.stackexchange.com/questions/3109896 – Vezen BU Mar 01 '23 at 12:57

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