# 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. \$\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 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 \$\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 AM
Human
Re: 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 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.