Upper bounds on the condition number of the eigenvector matrix

Question

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.

Answer 1

No, there are no such bounds.

Let $a>0$ be a fixed real and $\varepsilon\ll a$. Take the non-diagonalizable matrix $A=\begin{bmatrix}3a & 0 & 0\\ 0 & 0 & \varepsilon \\ 0 & 0 & 0 \end{bmatrix}$. Let $V$ be a symmetric, orthogonal matrix that maps $\begin{bmatrix}\sqrt {3}\\0\\0\end{bmatrix}$ to $\begin{bmatrix}1\\1\\1\end{bmatrix}$ (for instance, a Householder reflector).

Now $M=VAV^{-1}$ has entries in $[a-O(\varepsilon),a+O(\varepsilon)]$ and is not diagonalizable. So the eigenvector condition number of any sequence of matrices $M_k$ that converges to $M$ blows up.