The Frobenius, or Hilbert-Schmidt, 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 $\|A\|\_2$?



Absolute values and Frobenius norm
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 HS-norm 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. 


