Let N be an n*n matrix on a field F with nilpotency index n i.e. $\displaystyle N^n=0$ but $\displaystyle N^{n-1}\neq 0$ . Now show that N doesn't have square root i.e. there is no matrix like A where $\displaystyle A^2=N$.

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- Aug 6th 2013, 12:49 AMHumanA question on a nilpotent matrix
Let N be an n*n matrix on a field F with nilpotency index n i.e. $\displaystyle N^n=0$ but $\displaystyle N^{n-1}\neq 0$ . Now show that N doesn't have square root i.e. there is no matrix like A where $\displaystyle A^2=N$.

- Aug 6th 2013, 05:56 AMHallsofIvyRe: A question on a nilpotent matrix
Since the conclusion is

**negative**, that is, "there does**not**exist", an indirect proof is indicated. Suppose the conclusion where**false**. That is, suppose there exist A such that $\displaystyle A^2= N$. Then $\displaystyle A^{2n}= N^n= 0$. Now, what does that say about $\displaystyle A^n$? - Aug 6th 2013, 07:21 AMHumanRe: A question on a nilpotent matrix
$\displaystyle A^n=0$? And then for $\displaystyle n \geq 2$, $\displaystyle A^{2(n-1)}=0$ , a contradiction!

- Aug 7th 2013, 05:21 AMjohngRe: A question on a nilpotent matrix
Hi Human,

What you say is certainly true, but do you know why it follows that A^{n}=0 from A^{2n}=0? - Aug 7th 2013, 07:08 AMHumanRe: A question on a nilpotent matrix
If some power of A is zero then its nth power is zero.