When is the LU decomposition unique?

Question

I want to find out when a matrix decomposition $A = LU $ (L lower and U upper matrix) is unique? Clearly, if $A$ is not invertible, there is no chance that this decomposition is unique. Hence, assuming $A$ is invertible, then I would guess(!) that this decomposition is unique if it exists. But what can be said about $PA = LU$, where $P$ is a permutation matrix? Can we ever achieve uniqueness there? Of course, we have to assume $A$ being invertible here, too.