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Nilpotent Operator

  1. Mar 27, 2008 #1
    1. The problem statement, all variables and given/known data

    If T is a nilpotent transformation from V -> V, V - finite dimensional vector space.

    show that [tex]a_{0}+a_{1}T+\cdots+a_{k}T^{k}[/tex] is invertible. [tex]a_{0}[/tex] nonzero.

    Im having trouble finding the inverse, I know for 1+T+[tex]\cdots[/tex]+T[tex]^{m-1}[/tex]

    the inverse is (1-T),where T[tex]^{m}[/tex]=0. I also tried [tex]a_{0}^{-1}T^{m-1}[/tex]
    but this gives me [tex]T^{m-1}[/tex]
     
    Last edited: Mar 27, 2008
  2. jcsd
  3. Mar 27, 2008 #2

    Dick

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    Write that as -a0*I=a_1*T+a_2*T^2+...+a_k*T^k. Is that enough of a hint? If not notice the right side has a common factor.
     
  4. Mar 27, 2008 #3
    I see that it shows T to be invertible but how does it show that the original polynomial

    is invertible?
     
  5. Mar 27, 2008 #4

    Dick

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    Good point. I was solving the wrong problem.
     
  6. Mar 27, 2008 #5

    Hurkyl

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    Okay, so you figured to annihilate all of the terms of your polynomial except the T^{k-1} term. What else can you annihilate?
     
  7. Mar 27, 2008 #6

    Dick

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    I think you are going to have a hard time expressing an answer in closed form in terms of the a_i's. But think of it this way. If P is your polynomial, then P=a0+Q where Q is the rest of the terms. Can you show Q is nilpotent? Does that suggest how to invert a0+Q?
     
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