How to form the matrix representation of $|O|^3$ I'm interested in getting the matrix representation of the absolute value of an operator. I know the matrix representation of the operator $O$. Now how do I take its absolute value?
 A: You do the same thing you always do when applying a function to an observable - you diagonalize it, then apply the function to the eigenvalues, then undo the diagonalizing procedure.
The formal underpinning of this procedure is given by Borel functional calculus.
A: I assume you have not diagonalized the operator, else you would not be asking the question. The absolute value of the operator $\hat{O}$ is presumably $\sqrt{\hat{O}^\dagger \hat{O}}\equiv \hat{B} $, so the crucial question is what type of algorithm one would choose to evaluate the square root $\hat{B}$ of the operator $\hat{A}=  \hat{O}^\dagger \hat{O}=\hat{B} \hat{B}$.
If, for example, a suitable satisfactory series in  $\hat{A}$  were found for $\hat{B}$,  such as  $\hat{B}= \sqrt{I+(\hat{A}-I)}$ $= \sum_{n=0}^{\infty} {{1/2}\choose{n}} (\hat{A} -I )^n  $, etc. you'd be done.
Or else you might consider the $\tanh(k\hat{O})$ representation of the step function for large k, if easier to compute, so your quantity is $\tanh(k\hat{O}) ~\hat{O}^3$.
