Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Eigenvalues of a compact positive definite operator

  1. Dec 20, 2012 #1

    SVD

    User Avatar

    eigenvalues of a compact positive definite operator!!!

    Let A be a compact positive definite operator on Hilbert space H.
    Let ψ1,...ψn be an orthonormal set in H.
    How to show that <Aψ1,ψ1>+...+<Aψn,ψn> ≤ λ1(A)+...+λn(A), where
    λ1≥λ2≥λ3≥..... be the eigenvalues of A in decreasing order.
    Can someone give me a hint???
     
  2. jcsd
  3. Dec 20, 2012 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    Re: eigenvalues of a compact positive definite operator!!!

    Both the left and right expression look like tr(A).
     
  4. Dec 20, 2012 #3

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Re: eigenvalues of a compact positive definite operator!!!

    Try induction.

    Do you know that

    [tex]\lambda_1=sup\{<Ax,x>~\vert~x\in H,~\|x\|=1\}[/tex]

    ??

    If you know this, then the case n=1 should be easy. Can you find an argument to deal with the other cases?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Eigenvalues of a compact positive definite operator
Loading...