Let $A\in {\mathbb R}^{m\times n}_+$ be matrix with positive entries, and $B\in {\mathbb R}^{m\times n}$. Is it true: $$ \sigma_1(A\circ B)\leq \max_{i,j}a_{ij} \cdot \sigma_1(B) $$ In fact, I additionally have $A=[{\mathbf a}_1, \cdots, {\mathbf a}_n]$ and ${\mathbf a}_{i+1}\geq {\mathbf a}_{i}$ for $i=1,..,n-1$.
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1From this post, the answer is no if we don't have your additional condition. If we add the condition, I suspect that the answer is still no but I'm not sure. – Ben Grossmann Aug 19 '21 at 18:31