Let $A$ be an $n\times n$ real matrix with entries in a fixed interval $[a_\min,a_\max]$, with $a_\min$, $a_\max>0$.
Question: Are there any upper bounds on the condition number of the eigenvector matrix associated to $A$ as $n$ increases?
The motivation of this question comes from the $n\times n$ random dense case. Indeed, with reference to this case, in [1, Page 338] it is stated that: "[...] experiments indicate that the condition number of the associated eigenvector matrix also grows linearly [as $n$ increases]".
Hence, I wonder whether any similar result can be/has been proved for the deterministic (positive) dense case.
 L. N. Trefethen and M. Embree. Spectra and pseudospectra: the behavior of nonnormal matrices and operators. Princeton University Press, 2005.