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Homework Help: Stuck on step in a linear algebra proof

  1. Sep 4, 2011 #1
    1. The problem statement, all variables and given/known data
    Axler's Linear Algebra done right, propositions 7.6 states:
    An operator T is normal if and only if ||Tv|| = || T*v||, T* being the adjoint of T.

    I've been stuck on the last equivalence in the proof stated here:

    <T*Tv,v>=<TT*v> <=> ||Tv||^2=||T*v||^2, with squaring both sides of the second statement proves the theorem.

    2. Relevant equations
    3. The attempt at a solution
    Basically all my attempts have been stopped by the fact that i can't either get rid of T or T* in the second slot when working backwards or trying to get T or T* back when working through it forwards. Any help would be highly appreciated.
  2. jcsd
  3. Sep 4, 2011 #2


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    Homework Helper

    so i take it you definition of normal is:
    T*T = TT*

    and the problem you are having is:
    Assume ||Tv|| = || T*v||, show T is normal

    Now assuming all that is the case you squaring the relation is a simple matter
    ||T*v||^2 = ||Tv||^2

    Then writing in terms of inner products you have
    <T*v,T*v> = <Tv,Tv>

    now i would use the properties of operators in inner products to ove an operator within the inner product, so both act on a single vector within the inner product
    Last edited: Sep 4, 2011
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