1. Not finding help here? Sign up for a free 30min 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!

Norm question (Frobenius norm)

  1. Apr 10, 2010 #1
    We show that if P and Q are Hermitian positive definite matrices satisfying

    [tex]x^{*}Px \leq x^{*}Qx [/tex] for all [tex]x \in \textbf{C}^{n}[/tex]

    then [tex]\left\| P \right\|_{F} \leq \left\| Q \right\|_{F}[/tex]

    where [tex]\left\| \cdot \right\|_{F} [/tex] denotes the Frobenius norm (or Hilbert-Schmidt norm)

    If A is a mXn matrix, then the Frobenius norm of A is
    [tex]\left\| A \right\|_{F} = \left( \sum ^{m}_{i=1} \sum ^{n}_{j=1} \left| a_{ij}\right| ^{2} \right) ^{1/2} = \left( \sum ^{n}_{j=1} \left\| a_{j}\right\| ^{2}_{2} \right) ^{1/2} [/tex]

    with [tex] a_{j} [/tex] being the j-th column of A.

    I can see that the matrix (Q-P) is positive semi-definite. But from there I am not sure how to proceed.
    Last edited: Apr 10, 2010
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?

Similar Discussions: Norm question (Frobenius norm)
  1. Inf norm question (Replies: 0)