Absolute value of a matrix Let $A=(a_{ij})$ be an infinite matrix. Consider $|A|=(A^*A)^{1/2}$ and $A'=(|a_{ij}|)$. 
Is there any relation between $|A|$ and $A'$?  

The notation $|A|$ for $(A^*A)^{\frac{1}{2}}$ is due to an analogy of the polar decomposition of the matrix $A=U|A|$ where $U$ is a partial isometry to
the polar decomposition of a complex number $z=e^{i\arg(z)}|z|$.
There is no obvious connection to $A^{'}$.
The notation should not be interpreted as an absolute value. One does not have in general 
$|A+B|\leq |A|+|B|$ in the sense that the difference is a positive matrix.
If $A$ represents a Hilbert-Schmidt operator there is the following connection between $|A|$ and $A^{'}$:
$\sum_{1\leq ij<\infty}|a_{ij}|^2=\sum_{j=1}^{\infty}\sigma_j^2(A)$ 
where $\sigma_j(A)$ are the singular numbers of $A$, i.e. the eigenvalues of $|A|$.
