I have a square matrix $A_{n,n}$, and want to define it as $$ A = A' A'$$ where the RHS is a product of matrix $A'$ with itself. Is there an analytical way to deduce what $A'$ is, or even if it exists (other than trial and error)? I'm not sure if this is even possible, but am interested to see if there are any methods that exist.
Edit: My particular matrix $A$ may not be invertible so what further restrictions, if any, would this place on $A'$?