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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
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