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.