# Norm question (Frobenius norm)

1. Apr 10, 2010

### math8

We show that if P and Q are Hermitian positive definite matrices satisfying

$$x^{*}Px \leq x^{*}Qx$$ for all $$x \in \textbf{C}^{n}$$

then $$\left\| P \right\|_{F} \leq \left\| Q \right\|_{F}$$

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

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

with $$a_{j}$$ 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
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