If $A$ is normal and $A^{2006}=0$ then $A=0$

Asked Jun 23 '21 at 10:21

Active Jun 23 '21 at 10:21

Viewed 74 times

If $A$ is normal and $A^{2006}=0$ then $A=0$

since $A^{2006}=0$ the possible minimal polynomials of $A$ are $t^p, p\in\{1,2,\dots 2006\} $

My attempt: Suppose $A\ne 0$ then we get that $0$ is not a root of the minimal polynomial, but that's a contradiction since for all possible minimal polynomials $0$ is a root.

I don't know how to use the fact that $A$ is normal Any hints ?

asked Jun 23 '21 at 10:21

領域展開

2,631

1

Possible duplicate of https://math.stackexchange.com/questions/68762/a-is-normal-and-nilpotent-show-a-0 – lhf Jun 23 '21 at 10:35
If $A=\begin{bmatrix}1&1\1&1\end{bmatrix}$ the minimal polynomial is $A^2-2A$, which has $0$ as a root. – David C. Ullrich Jun 23 '21 at 11:59

If $A$ is normal and $A^{2006}=0$ then $A=0$

0 Answers0