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: 2-norm math help

  1. Apr 7, 2010 #1
    How do you prove that

    [tex]\left\|A^{*} A \right\|_{2}= \left\| A \right\|^{2}_{2}[/tex] ?

    I can prove that [tex]\left\|A^{*} A \right\|_{2} \leq \left\| A \right\|^{2}_{2}[/tex]

    but I am not sure how to do it for the other inequality.
     
  2. jcsd
  3. Apr 7, 2010 #2

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Re: 2-norm

    What is A and what is *? For example, it isn't true for 2x2 matrices.
     
  4. Apr 7, 2010 #3
    Re: 2-norm

    I am sorry, let me specify: [tex]A \in \textbf{C}^{m\times n}[/tex] and [tex]A^{*}[/tex] is the conjugate transpose of A.

    Is it true that [tex]\left\| A^{*}A \right\|_{2} \geq \left\| A \right\|^{2}_{2} [/tex] ?

    If yes, how do you prove this
     
  5. Apr 7, 2010 #4

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Re: 2-norm

    Try it with

    A = [[1,2],[3,4]]

    Edit: Better yet, try the identity matrix.
     
    Last edited: Apr 7, 2010
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook