Superdiagonal for the Jordan form of a Jordan block power

Question

The question is an extension of the Prove that $A$ is similar to $A^n$ based on A's Jordan form.

Let $J$ be Jordan block of any form.
In what circumstances Jordan form of power $J^n$ has the same superdiagonal as $J$?

Answer 1

Let $m$ be the size of the block. Then $J = \lambda + N$ with $N^{m-1}\neq 0$ and $N^m=0$.

For $\lambda = 0$ the answer is clearly negative.

For $\lambda \neq 0$, you can just extend the given proof:

$$J^n = \lambda^n + n\lambda^{n-1}N + \dotsb + \binom{n}{m-1}N^{m-1},$$ hence $(J^n-\lambda^n)^{m-1} = (n\lambda^{n-1}N)^{m-1} \neq 0$, i.e. the Jordan form of $J^n$ is given by one Jordan block of size $m$, i.e. the superdiagonal is the same as for $J$, namely $N$.