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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 $U$? ($A=LU$)

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Note that an $LU$ decomposition does not always exist: for instance among permutation matrices, only the identity matrices have an $LU$ decomposition. If an $LU$ decomposition exists it can be made unique by requiring diagonal entries $1$ for (say) $L$; however if a permutation matrix is thrown into the mix, (e.g. $LUP$) then uniqueness is no longer possible. The general notion is Bruhat decomposition, which at the element level is not unique. – Marc van Leeuwen Aug 21 '12 at 12:34

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.

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It seems you're assuming a certain dimension? In any event: one often uses the degrees of freedom available to impose constraints on either the upper or lower triangular factor, e.g. having one of them be unit triangular... – J. M. Aug 21 '12 at 10:31
Edited! For some reason I had $3\times3$ matrices in my brain... – Daryl Aug 21 '12 at 12:13

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