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Hermitian Transformations

  1. Mar 1, 2006 #1
    This is the problem:

    Let T be a complex linear space with a complex inner product <.,.>. Define T in L(V,V) to be Hermitian if <Tv,v> = <v,Tv> for all v in V.
    Show that T is Hermitian iff <Tv,w> = <v,Tw> for all v,w in V [Hint: apply the definition to v+w and to v+iw].

    So this was my thought process:
    <T(v+w),v+w> = <v+w,T(v+w)>
    <Tv+Tw,v+w> = <v+w,Tv+Tw>
    <Tv,v> + <Tv,w> + <Tw,v> + <Tw,w> = <v,Tv> + <v,Tw> + <w,Tv> + <w,Tw>
    And the terms with the same variables cancel out by definition so this leaves
    <Tv,w> + <Tw,v> = <v,Tw> + <w,Tv>
    which doesn't really help.

    How do I go about doing this? Thanks.
  2. jcsd
  3. Mar 1, 2006 #2


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    Follow the entire hint.
  4. Mar 1, 2006 #3
    I feel really dumb for asking this, but what? Simultaneously? I'm so confused
  5. Mar 1, 2006 #4
    I think what he's getting at is to do the "v+iw" part and compare the two answers.

  6. Mar 1, 2006 #5


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    Simply add the 2 results and then simplify by 2.

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