Nilpotent matrices and nilpotent level

Question

I would like to know what is a nilpotent matrix and nilpotency level? Are these matrices invertible?

Answer 1

A matrix is nilpotent if $A^n$ is zero for some $n \ge 1$. They are never invertible since their eigenvalues are all $0$. The elements of such matrix on the first upper subdiagonal i.e. $a_{i,i+1}$ are considered to be level one. The next subdiagonal is level 2 and so on.

Answer 2

A matrix $A$ is nilpotent iff some power of it is zero, that is if there is a $k \in \mathbb N$ with $A^k = 0$. The smallest such $k$ is called the nilpotency degree of $A$ (you call this level of nilpotency?).

If $A$ were nilpotent and invertible, we would have for $k$ with $A^k=0$ that \[ \mathrm{Id} = (AA^{-1})^k = A^kA^{-k} = 0 \] which is absurd.