Let $A$ and $B$ be two $n\times n$ random matrices. Matrix $A$ has coefficients taken from a normal distribution $ \mathcal{N}(\mu_A,\sigma_A)$, and matrix $B$ has coefficients taken from $ \mathcal{N}(\mu_B,\sigma_B)$. Can we identify a non-trivial upper bound for the expected maximum coefficient of the product $A\cdot B$?
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2$\begingroup$ What would count as "trivial"? That is, what's the "trivial" upper bound that you want to improve on? $\endgroup$– Noam D. ElkiesCommented Aug 30, 2022 at 21:12
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$\begingroup$ A trivial upper bound would be for example the larget absolute cofficient value in A multiplied by the largest absolute cofficient value in B multiplied by n $\endgroup$– Ron BannerCommented Aug 31, 2022 at 4:56
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