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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 $A'$?

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    Do you really mean _infinite_ (or maybe _indefinite_)? While in some context infinite matrices may be considered, it does not look like this is one. – 2017-01-26
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    You call determinant the absolute 'value of a matrix'? – 2017-01-26
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    @MarcvanLeeuwen : I mean infinite matrice – 2017-01-26
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    If I am right, the Eigenvalues of $|A|$ are the absolute values of those of $A$, and have nothing to do with those of $A'$. – 2017-01-26
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    @niki I found a connection in the case of Hilbert-Schmidt operators and added it in the answer, in this case the sum $\sum_j |Ae_j|^2=\sum_{ij}|(Ae_i,e_j)|^2=\sum_{ij}|a_{ij}^2$ does not depend on the choice of the basis – 2017-01-26
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    @niki Then you should state more context. What every you mean by $A^*$, I am pretty sure $A^* A$ is not defined for all infinite matrices (take for instance one with all entries equal to $1$). – 2017-01-26
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    @Marc van Leeuwen I think what is ment is the adjoint of the operator represented by the matrix, that always exists by Riesz´s representation theorem – 2017-01-26
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    @PeterMelech The problem is not the adjoint, it is the multiplication of infinite matrices. – 2017-01-26
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    @Marc van Leeuwen I´m sorry. Of course You´re right,it only makes sense if the matrix can be interpreted as the representation of a bounded linear map! – 2017-01-26

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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 $|A|$.