1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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

    User Avatar
    Science Advisor
    Education Advisor

    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:
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted