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.

**[1]** L. N. Trefethen and M. Embree. *Spectra and pseudospectra: the behavior of nonnormal matrices and operators*. Princeton University Press, 2005.