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Monic polynomial problem

  1. Dec 15, 2008 #1
    1. The problem statement, all variables and given/known data

    Let us say that p(c) is a monic polynomial such that when applied to a particular v, we have pv=0. Let V be a finite dimensional vector space. Let V be the direct sum of k invariant subspaces. Then v = v_1+...+v_k.

    When I apply pv=0 does this imply that pv_1=pv_2=0.

    2. Relevant equations



    3. The attempt at a solution
    p(z)=a_0+a_1z+...a_mz^m
    p(T)v=a_0+a_1(T)(v_1+v_2..v_m)+...a_m(T^m)(v_1+v_2..v_m)
    We know that a_0,a_1 are not all 0s.If they are then pv = 0 for all v. So we must have that a_0 = -(the rest) or the terms must cancel out each other.
    or we have (T)(v_1+v_2..v_m)=0 hence T^m(v_1+v_2..v_m)=0.

    This is where I am stuck
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Dec 15, 2008 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Re: polynomial

    What do you mean by "applying a polynomial to a vector"?
     
  4. Dec 15, 2008 #3
    Re: polynomial

    I mean pv=0 for a particular v
     
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