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Math Help - Nilpotent operator

  1. #1
    MHF Contributor Also sprach Zarathustra's Avatar
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    Nilpotent operator

    Let T:V\to V be linear transformation over field, and let p be a polynomial over that field.
    Prove that p(T) nilpotent iff p(0)=0


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    Quote Originally Posted by Also sprach Zarathustra View Post
    Let T:V\to V be linear transformation over field, and let p be a polynomial over that field.
    Prove that p(T) nilpotent iff p(0)=0


    Thanks!
    Since p is a polynomial, we can write p(x) = \sum_{i=0}^n a_ix^i. Now, note that p(0) = a_0 , ~ p(0) = 0 \Rightarrow a_0 = 0. Conclude by using the fact that T is nilpotent.
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    MHF Contributor Also sprach Zarathustra's Avatar
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    Quote Originally Posted by Defunkt View Post
    Since p is a polynomial, we can write p(x) = \sum_{i=0}^n a_ix^i. Now, note that p(0) = a_0 , ~ p(0) = 0 \Rightarrow a_0 = 0. Conclude by using the fact that T is nilpotent.
    Yes, I'm aware this fact.

    Check me please...

    i) [LaTeX ERROR: Convert failed] nilpotent.
    [LaTeX ERROR: Convert failed]
    Above, sum of nilpotent operators [LaTeX ERROR: Convert failed] nilpotent.

    ii) [LaTeX ERROR: Convert failed]
    So, there exist [LaTeX ERROR: Convert failed] , index of nilpotents so that[LaTeX ERROR: Convert failed]
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  4. #4
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    Quote Originally Posted by Also sprach Zarathustra View Post
    Yes, I'm aware this fact.

    Check me please...

    i) [LaTeX ERROR: Convert failed] nilpotent.
    [LaTeX ERROR: Convert failed]
    Above, sum of nilpotent operators [LaTeX ERROR: Convert failed] nilpotent.
    This is good.

    ii) [LaTeX ERROR: Convert failed]
    So, there exist [LaTeX ERROR: Convert failed] , index of nilpotents so that[LaTeX ERROR: Convert failed]
    This needs some justifications (simple), and otherwise you are done.
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  5. #5
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    Quote Originally Posted by Also sprach Zarathustra View Post
    i) p(0)=0 \implies p(T) nilpotent.
    p(T)=a_1T+a_2T^2+ \dots +T^n
    Above, sum of nilpotent operators \implies p(T) nilpotent.
    Maybe you should say "sum of commuting nilpotent operators \implies p(T) nilpotent." It's not true in general that a sum of nilpotent operators is nilpotent.
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