I believe that the absolute value of a matrix is defined as $$ |A|=\sqrt{A^{\dagger}A} \ . $$ But the square root of a matrix is not unique wikipedia gives a list of examples to illustrate this.
To understand this, how does one work out the absolute value of: $$ A=\begin{pmatrix}1 & 0\\0 & -1\end{pmatrix} $$ Clearly $A^{\dagger}=A$ so $|A|=\sqrt{A^2}$, but this is not necessarily $A$. I want to pick the identity in this case, since then the eigenvalues of $|A|$ are both 1 (and they were $\pm1$ for $A$). But mathematics is not about what I want. So what is $|A|$? Is it well-defined? And how do I do this operation in general, since my application for this is of course far more complex.
If $D$ is a diagonal matrix with positive terms, then the positive square root, $\sqrt D$ is uniquely determined by the diagonal matrix of positive square roots of diagonal terms.
If a matrix $A$ is diagonalizable with positive eigenvalues then $A= P^{-1}DP$ and we can define its positive square root as $ \sqrt A= P^{-1} \sqrt D P$
Thus there is no confusion in finding the absolute value if $A$ if we consider only positive square roots.