2x2-matrix/Nilpotent/Jordan/Example

A matrix of the form

${\displaystyle {}M={\begin{pmatrix}0&a\\0&0\end{pmatrix}}\,}$

with ${\displaystyle {}a\neq 0}$ has, with respect the basis ${\displaystyle {}ae_{1}}$ and ${\displaystyle {}e_{2}}$, the form

${\displaystyle {\begin{pmatrix}0&1\\0&0\end{pmatrix}}.}$