what is the difference between the jordan form of matrices A
and diagonal block matrices similar to A
?
Difference, in what terms? For example
$\displaystyle A\sim J=\begin{bmatrix}{\lambda}&{1}&{0}\\{0}&{\lambda}& {1}\\{0}&{0}&{\lambda}\end{bmatrix}\Rightarrow A\sim K=\begin{bmatrix}{\lambda}&{2}&{0}\\{0}&{\lambda}& {3}\\{0}&{0}&{\lambda}\end{bmatrix}$
$\displaystyle J$ is the Jordan form of $\displaystyle A$, $\displaystyle K$ is a diagonal block matrix similar to $\displaystyle A$ but it is not its Jordan form.