Gauss decomposition of a matrix is also called LU decomposition. Let $A$ be a matrix. Then $A=LU$ for some lower triangular matrix $L$ and upper triangular matrix $U$. This can be obtained using Gauss algorithm.
It is said that Bruhat decomposition in $GL_n$ case is the Gauss decomposition. Why Bruhat decomposition in $GL_n$ case is the Gauss decomposition? Thank you very much.
In the case of $GL_2$. We have Bruhat decomposition $GL_2 = \cup_{w \in W} BwB $, where $W$ is the Weyl group, $B$ is the subgroup of $GL_2$ consisting of all upper triangular matrices.