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Nilpotent Matrices

  1. Sep 11, 2012 #1
    1. The problem statement, all variables and given/known data
    Suppose that N is a nilpotent mxm matrix, N[itex]^{m}[/itex]=0, but N[itex]^{m'}[/itex][itex]\neq[/itex]0 for m'<m. Show that there exists a basis in which it takes the form of a single Jordan block with vanishing diagonal elements. Prove that your basis set is linearly independent.


    2. Relevant equations



    3. The attempt at a solution
    So I've recognized the fact that N[itex]^{m'}[/itex][itex]\neq[/itex]0 for m'<m means that N[itex]^{m'}[/itex] does not annihilate every vector in V. I'm just not really sure where go from here...
     
  2. jcsd
  3. Sep 11, 2012 #2

    Dick

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    Pick a basis for the vectors that are annihilated by N. Then add a basis for the vectors that are annihilated by N^2 but not by N. Continue. Eventually you'll get a basis for the whole space, right? What does N look like in that basis?
     
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