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Homework Help: Jordan canonical form 1

  1. May 15, 2010 #1
    1. The problem statement, all variables and given/known data
    Let T:V->W be a linear transformation. Prove that if V=W (So that T is linear operator on V) and λ is an eigenvalue on T, then for any positive integer K
    N((T-λI)^k) = N((λI-T)^k)


    2. Relevant equations
    T(-v) = -T(v)
    N(T) = {v in V: T(v)=0} in V hence T(v) = 0 for all v in V.


    3. The attempt at a solution
    we know that (T-λI)^k(-v) = -(T-λI)^k(v) = (λI-T)^k(v). So when (T-λI)^k(v) = 0 so does (λI-T)^k(v). Hence N((T-λI)^k) = N((λI-T)^k)
     
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  3. May 16, 2010 #2

    Landau

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    You mean Tv=0 for all V in N.
    You mean (T-λI)^k(v) = 0 if and only if (λI-T)^k(v) = 0.
     
  4. May 16, 2010 #3
    Yeah, that is what I am trying to prove.
     
  5. May 16, 2010 #4

    Landau

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    It looks correct. Basically, you're proving that for any linear map T, T and -T have the same kernel. This is true because Tv=0 iff -Tv=0.
     
  6. May 16, 2010 #5
    Can we just state that because Tv=0 iff and -Tv = 0 therefore N((T-λI)^k) = N((λI-T)^k) or do we have to prove that Tv=0 iff and -Tv = 0?
     
  7. May 17, 2010 #6

    Landau

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    Well, there is not much to prove. The only thing you need is that -0=0.
     
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