Proof of uniqueness of LU factorization The first question: what is the proof that LU factorization of matrix is unique? Or am I mistaken?
The second question is, how can theentries of L below the main diagonal be obtained from the matrix $A$ and $A_1$ that results from the row echelon reduction to $U$? ($A=LU$) 
 A: It is unique if you require $diag(L) = (1, \dots 1)$. 
Because, assume that the LU-decomposition is not unique. Any other LU-decomp would be of the form:
$$A=LU=LI_nU=LDD^{-1}U=(LD)(D^{-1}U)$$ with $D\in\mathbb{R}^{n\times n}$, $D\neq I_n$. For this $(LD)(D^{-1}U)$ to be a LU-decomposition, we need $LD$ lower triangular, $(D^{-1}U)$ upper triangular. In order for $LD$ to stay a lower triangular matrix, $D$ needs to be lower triangular. But $D$ lower triangular $\Rightarrow D^{-1}$ lower triangular matrix $\Rightarrow$ $(D^{-1}U)$ not upper triangular. 
Therefore, $D$ can only be a diagonal matrix. 
Now, because $D\neq I_n \Rightarrow \exists d_{ii} \neq 1$ but in that case $$ (LD)_{ii} = \sum_{j=1} L_{ij}D_{ij} = L_{ii}D_{ii} + \sum_{j=1, j=i} L_{ij}*0 = 1\times D_{ii} +0 \neq 1$$ 
Resulting in $diag(LD) \neq (1, \dots 1)$ which is a contradiction. So such a $D$ does not exist, meaning the decomposition is unique. 
A: The factorisation is not unique. There are $n^2+n$ coefficients to estimate and only $n^2$ "equations". As such, that is why there are the two "common" methods, Doolittle and Crout see wiki page. For each of these two approaches, you can show that the resulting linear system has a unique solution.
