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Math Help - T is a unitary operator. Prove |<Tv,v>| is less than or equal to <v,v>

  1. #1
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    T is a unitary operator. Prove |<Tv,v>| is less than or equal to <v,v>

    Let V be an inner product space of finite dimension over C (complex numbers). Let T be a unitary linear operator from V to V. Prove that:
    |<Tv,v>| is less than or equal to <v,v>

    My working so far:
    Since T is unitary <Tv,Tv>=<v,v>, but I'm not sure if this helps at all since I don't know how to show that |<Tv,v>| <= <Tv,Tv>.

    Thanks
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  2. #2
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    Re: T is a unitary operator. Prove |<Tv,v>| is less than or equal to <v,v>

    I got the answer:
    According to the Cauchy Schwartz inequality:
    |<Tv,v>| <= ||Tv||.||v||
    T is unitary so:
    ||Tv||=||v||
    So:
    |<Tv,v>| <= ||Tv||.||v||=||v||.||v||=<v,v>
    QED
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