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Invariance of quadratic form for unitary matrices

  1. Nov 6, 2015 #1
    1. The problem statement, all variables and given/known data

    Show that all ##n \times n## unitary matrices ##U## leave invariant the quadratic form ##|x_{1}|^{2} + |x_{2}|^{2} + \cdots + |x_{n}|^{2}##, that is, that if ##x'=Ux##, then ##|x'|^{2}=|x|^{2}##.

    2. Relevant equations

    3. The attempt at a solution

    ##|x'|^{2} = (x')^{\dagger}(x') = (Ux)^{\dagger}(Ux) = x^{\dagger}U^{\dagger}Ux = x^{\dagger}x = x^{2}##.

    Am I correct?
     
  2. jcsd
  3. Nov 6, 2015 #2

    Krylov

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    Yes. (Although I always need to get used to the physicist's notation for the Hermitian transpose. :wink:)
     
  4. Nov 6, 2015 #3
    Thanks!

    I always wish I could see from the mathematician's point of view, being as I am from a Physics background. :smile:
     
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