Matrix/Nilpotent/Typical form/Example

Let ${}M$ be an upper triangular matrix with the property that all diagonal entries are ${}0$. Thus, ${}M$ has the form

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

Then ${}M$ is nilpotent, with every power the ${}0$-diagonal is moved one step up and to the right. If, for example, the product of the ${}i$-th row and the ${}j$-th column with

${}i\geq j-1\,$

is computed, then there is always a ${}0$ in the partial products and, altogether, the result is ${}0$.