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Math Help - Prove if T is Hermitian, then <T(x), x> is real for alll x in V.

  1. #1
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    Prove if T is Hermitian, then <T(x), x> is real for alll x in V.

    Let T be a linear operator on a complex inner product space V.

    1.Prove if T is Hermitian, then <T(x), x> is real for alll x in V.

    2. Prove that if <T(x), x> is real for all x in V, then T is Hermitian.


    I'd be grateful with any degree of help!

    thanks!
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  2. #2
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    Quote Originally Posted by ericmik View Post
    Let T be a linear operator on a complex inner product space V.

    1.Prove if T is Hermitian, then <T(x), x> is real for alll x in V.

    2. Prove that if <T(x), x> is real for all x in V, then T is Hermitian.


    I'd be grateful with any degree of help!

    thanks!

    i sorta came up with a solution here,

    T = T^*

    then<T(x), x> = <x, T(x)>
    by the definition of inner product space: <T(x), x> = conjugate(<x, T(x)>)

    thus <x, T(x)> = conjugate<x, T(x)>

    but is it sufficient to say <x, T(x)> is real??
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  3. #3
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    Quote Originally Posted by ericmik View Post
    i sorta came up with a solution here,

    T = T^*

    then<T(x), x> = <x, T(x)>
    by the definition of inner product space: <T(x), x> = conjugate(<x, T(x)>)

    thus <x, T(x)> = conjugate<x, T(x)>

    but is it sufficient to say <x, T(x)> is real??
    Consider <x,T(x)>=Re(<x,T(x)>)+i*Im(<x,T(x)>)
    Conjugate of <x,T(x)>=Re(<x,T(x)>)-i*Im(<x,T(x)>)
    Now what if you add these two equations ?
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