Bruhat decomposition

Bruhat decomposition is the name for the fact that $B\backslash \backslash G/B=W$, where $G$ is a reductive group, $B$ a Borel subgroup, and $W$ the Weyl group. Less canonically, one can write $G=BWB$.

In the case of the general linear group $G=G{L}_n$, $B$ is the group of nonsingular upper triangular matrices, and $W$ is the collection of permutation matrices (and is isomorphic to $S_n$). Any nonsingular matrix can thus be written uniquely as a product of an upper triangular matrix, a permutation matrix, and another upper triangular matrix.