A standard form of LU factorization

Question

Why LU factorization of a matrix is not unique?

Is there a standard form of LU factorization of a matrix or different forms are acceptable ?

Answer 1

Just a simple example that pops into my mind. Let $A\in \mathbb{R}^{n\times n}$ and $B \in \mathbb{R}^{n\times n}$. Assume $A$ is the zero matrix. Then any $B$ can be used for the LU decomposition of the zero matrix.