I'm wondering whether there is certain relationship between the largest eigenvalue of a positive matrix(every element is positive, not neccesarily positive definite) $A$, $\rho(A)$ and that of $A∘A^T$, $\rho( A∘A^T)$, where $∘$ denotes hadamard product.

Here's a result I find for many numerical cases. I create a matrix of size $n$ whose elements are uniformly drawn from $[0,M]$, as $n$ gets large (>20), $\rho(A)\rightarrow 2M\rho( A∘A^T)$.

I've read some papers on the bound of eigenvalue of $A∘B$, yet none of them mention the special case of $A∘A^T$. I'm wondering whether there's a theory about this and moreover, whether this result could be extended to general linear operators, such as integral operators $T(f(x))=\int k(x,y)f(y)dy$ and $T(f(x))=\int k(x,y)k(y,x)f(y)dy$

Any reference is appreciated. hanks in advance!

Bounds on eigenvalues of the Hadamard product, it may have something to do with the largest diagonal element of $A$ $\endgroup$ – Sylvan Jun 17 '14 at 17:49