Matrix/Nilpotent/Total Jordan block/Basis vectors/Example

A special case of example is the matrix

${\begin{pmatrix}0&1&0&\cdots &\cdots &0\\0&0&1&0&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&0&1&0\\0&\cdots &\cdots &0&0&1\\0&\cdots &\cdots &\cdots &0&0\end{pmatrix}}.$

An important observation is that under this mapping, ${}e_{n}$ is sent to ${}e_{n-1}$, ${}e_{n-1}$ is sent to ${}e_{n-2}$, and , finally, ${}e_{2}$ is sent to ${}e_{1}$, which is sent to ${}0$. The ${}(n-1)$-th power of the matrix sends ${}e_{n}$ to ${}e_{1}$ and is not the zero matrix, but the ${}n$-th power of the matrix is the zero matrix.