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.
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See also http://math.stackexchange.com/questions/184936/proof-of-uniqueness-of-lu-factorization and http://math.stackexchange.com/questions/445457/lu-decomposition-that-is-not-unique. – lhf Jul 10 '14 at 13:31
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See http://en.wikipedia.org/wiki/LU_decomposition#Existence_and_uniqueness. – lhf Jul 10 '14 at 13:32
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1@lhf so no uniqueness for $PA = LU$ in general, right? – Jul 10 '14 at 13:34
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It depends what "in general" means. It is unique if an $LU$ factorisation of a nonsingular $PA$ exists for some permutation matrix $P$ and if you fix in some sense the diagonal elements of $L$ and/or $U$, e.g., by requiring that $L$ has ones on the diagonal. – Algebraic Pavel Jul 11 '14 at 11:14