# A question on a nilpotent matrix

• Aug 6th 2013, 12:49 AM
Human
A question on a nilpotent matrix
Let N be an n*n matrix on a field F with nilpotency index n i.e. $N^n=0$ but $N^{n-1}\neq 0$ . Now show that N doesn't have square root i.e. there is no matrix like A where $A^2=N$.
• Aug 6th 2013, 05:56 AM
HallsofIvy
Re: 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 $A^2= N$. Then $A^{2n}= N^n= 0$. Now, what does that say about $A^n$?
• Aug 6th 2013, 07:21 AM
Human
Re: A question on a nilpotent matrix
$A^n=0$? And then for $n \geq 2$, $A^{2(n-1)}=0$ , a contradiction!
• Aug 7th 2013, 05:21 AM
johng
Re: A question on a nilpotent matrix
Hi Human,
What you say is certainly true, but do you know why it follows that An=0 from A2n=0?
• Aug 7th 2013, 07:08 AM
Human
Re: A question on a nilpotent matrix
If some power of A is zero then its nth power is zero.