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Math Help - A question on a nilpotent matrix

  1. #1
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    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.
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  2. #2
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    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?
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  3. #3
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    Re: A question on a nilpotent matrix

    A^n=0? And then for n \geq 2, A^{2(n-1)}=0 , a contradiction!
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    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?
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  5. #5
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    Re: A question on a nilpotent matrix

    If some power of A is zero then its nth power is zero.
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