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Math Help - matrix nilpotencies

  1. #1
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    matrix nilpotencies

    Hi everyone. Do you know if any matrix exists with a nilpotency of 0?
    A nilpotent matrix N is a square matrix defined as N^k=0 for k>=0....but isn't any square matrix to the zero power the identity matrix....so there is no nilpotent matrix with nilpotency 0??
    is my thinking correct?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: matrix nilpotencies

    Quote Originally Posted by cp05 View Post
    ....so there is no nilpotent matrix with nilpotency 0?? is my thinking correct?
    Yes, you are right.
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  3. #3
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    Re: matrix nilpotencies

    yes. nilpotency is only defined for positive powers of k. A^0 is by definition the identity matrix, for any non-zero matrix A. the zero matrix is a special case, just like the number 0^0 is a special case, although most sources (wolframalpha/mathematica, for example) define it as I (which, oddly enough, is a conflict when one considers 1x1 matrices).
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