The Frobenius, or HilbertSchmidt, norm of an $n$ by $n$ matrix $A$ is defined as $\A\_2 = \sqrt{\sum_{i,j=1}^n A_{ij}^2}$. The absolute value of $A$ is the unique positive matrix $A$ satisfying $A^2 = A^* A$. Are there any known relations between $\ A \_2$ and $\A\_2$?
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closed as offtopic by Ricardo Andrade, Daniel Moskovich, Chris Godsil, Jack Huizenga, Suvrit Dec 1 '13 at 6:13
This question appears to be offtopic. The users who voted to close gave these specific reasons:
 "This question does not appear to be about research level mathematics within the scope defined in the help center." – Daniel Moskovich, Chris Godsil, Suvrit
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It holds that A_2 =A_2 Proof: Let the singular value decomposition of A be given by $A=U\Sigma V^H$. Since the HSnorm is invariant under unitary transformations, $A_2= \Sigma_2$ holds. As for $A$, we obtain $A= A^HA=V\Sigma U^HU\Sigma V^H=V\Sigma^2 V^H= V\Sigma V^H$. Again, unitary invariance yields $ A _2=\Sigma _2=A_2$, qed. 

