I'm wondering about if there is some way to define, uniquely or not a canonical form which has minimal radii for Gersgórin discs. To be more specific for a given matrix $\bf A$, find $\bf C$ and $\bf T$:
$${\bf A = TCT}^{-1}, \text{with } {\bf C} \text{ sparse canonical structure, and minimal Gershgórin radii}$$
Obviously if $\bf A$ diagonalizable we would have a minimum at it's diagonalization, as all radii = 0. But in the general case, how could one argue to find a canonical form that minimizes the radii in some sense?
Also, I am not sure how to define minimal radii since increasing one radius could decrease another, so some kind of weighted average, maybe a $k$-norm on all non-diagonal elements would be reasonable? $$\sqrt[k]{\sum_{i\neq j}|{\bf C}_{ij}|^k} \text{ or } \sqrt{\sum_j\|{\bf c}_{j}\|_k}, \text{ where } {\bf c}_k \text{ is kth column vector stripped of its diagonal}$$
Maybe the choice of $k$ affects which form would be suitable. Sorry I realize it became a rather fuzzy question. I'd be happy for any feedback really. Could remake a better question later if requested.
I think this is probably not possible in general. For a simple example, the matrix
$$ \left(\begin{array}{cc}1 &1 \\ 0 & 1\end{array}\right) $$
has conjugates
$$ \left(\begin{array}{cc}1 &x \\ 0 & 1\end{array}\right) $$
for $x \neq 0$. So you can make its Gershgorin radii arbitrarily small, but not zero...