I know that the off-diagonals of the Jordan blocks for a matrix can be made arbitrarily small, i.e. we can find a similarity transformation such that the Jordan blocks look like this (as an example):
$$ J = \begin{bmatrix} \lambda & \delta & 0 \\ 0& \lambda & \delta \\ 0 & 0 & \lambda \end{bmatrix} $$
where $\delta > 0$ is arbitrarily small. In fact it is quite easy to show by using the construction of the generalized eigenvectors as follows:
$$\begin{align} (A - \lambda I) x_1 &= 0 \\ (A - \lambda I) x_2 &= \delta x_1 \\ (A - \lambda I) x_3 &= \delta x_2 \end{align}$$
which yields
$$ A \begin{bmatrix} x_1 & x_2 & x_3 \end{bmatrix} = \begin{bmatrix} x_1 & x_2 & x_3 \end{bmatrix} \begin{bmatrix} \lambda & \delta & 0 \\ 0& \lambda & \delta \\ 0 & 0 & \lambda \end{bmatrix} $$
However, I couldn't find any reference that mentions this "well-known" fact. Can you show a reference that mentions this?