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: Normal operators with real eigenvalues are self-adjoint

  1. Oct 11, 2009 #1
    Prove that a normal operator with real eigenvalues is self-adjoint

    Seems like a simple proof, but I can't seem to get it.

    My attempt: We know that a normal operator can be diagonalized, and has a complete orthonormal set of eigenvectors.

    Let A be normal. Then A= UDU* for some diagonal matrix D and unitary U. Also, A*=U*D*U

    Since D is the diagonal matrix of the eigen values of A, D is real, and thus D=D*.

    Thus D=U*AU = UA*U*.

    Then, I just get stuck on A=U²A*(U*)².
  2. jcsd
  3. Oct 11, 2009 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    You seem to be using the substitution
    (ST)* --> S*T*,​
    but the left hand side is not equal to the righthand side; making this substitution is not a valid deduction.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook