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The wikipedia page on Lie groups decompositions reads:

The LU decomposition of a dense subset in the general linear group. It can be considered as a special case of the Bruhat decomposition.

I am struggling to find a proof of the fact that the matrices which have an LU decomposition without pivoting are dense in $GL(n)$.

I would also be glad if someone could point me to a good reference for factorisations on Lie groups, in particular this link between the LU decomposition and the Bruhat decomposition.

Lezkus
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    Your first question (density) follows from Corollary 3 in https://math.stackexchange.com/questions/2951461/lu-factorization-of-a-nonsingular-matrix-exists-if-and-only-if-all-leading-princ/2955663#2955663 . No idea about references. – darij grinberg Sep 08 '19 at 01:09
  • I also found that the first question and its relation to the second follow from the following two posts: https://math.stackexchange.com/questions/637921/why-bruhat-decomposition-in-gl-n-case-is-the-gauss-decomposition https://math.stackexchange.com/questions/784954/conditions-for-a-matrix-in-bw-0b?rq=1 Thank you for your answer as well! – Lezkus Sep 08 '19 at 10:54

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