Say I have a $2 \times 2$ matrix $M$ with the property that $$||M_{ij} - M_{i'j'}|| \geq k$$ for some constant $k$ and all $i, j, i', j' \in \{1, 2\}$. This would imply that the entries of $M$ are sufficiently different from one another.
My question is whether these pairwise distances imply anything about the smallest singular value of $M$. Intuitively, it would seem that the distance lower bound between entries would imply that the rows are very different and, thus, far from being linearly independent and having a large $\sigma_2$. Beyond intuition though, is there actually a way to lower bound the smallest eigenvalue given only a constraint on pairwise distances and no additional structure?