nLab Gauss decomposition




Let kk be a field and GM n(k)G \in M_n(k) am n×nn \times n square matrix with entries in kk.

Then a Gauss decompositions of GG is a decomposition as G=UDLG = U D L where:

  1. LL is a lower triangular unidiagonal matrix,

  2. UU is upper triangular unidiagonal

  3. DD is diagonal.

Here unidiagonal means with only unit entries on the diagonal.

(If DD is the identity matrix then this is a special case of an LU decomposition.)

The notion of Gauss decomposition may be generalized to matrices entries in a noncommutative ring, in which case there are explicit formulas involving quasideterminants and, in the case of quantum linear groups, in terms of quantum minors. See at quantum Gauss decomposition for more.



Not every matrix has a Gauss decomposition: one may try to find a solution via the Gauss elimination procedure, and the conditions for the solution involve non-vanishing of certain “principal” minors of the matrix. Then the solution for entries of UU or LL involves ratios of (determinants of) minors of the same size.

For invertible matrices

If GG is an invertible matrix, GG \in GL n ( k ) GL_n(k) , then there always exist an n×nn \times n permutation matrix matrix ww such that a decomposition of the form G=wUDLG = w U D L exists.

The subset of matrices GGL n(k)G \in GL_n(k) for which such a decomposition exists (and then it is automatically unique and given by universal formulas) is a Zariski open subset in GL n(k)GL_n(k), called a “Gauss cell”.

The decomposition of matrices for any ww is also at times called a Gauss decomposition, in which case for w=1w=1 one speaks of the main cell.

The group GL n(k)GL_n(k) is covered by n ! n! such open subsets, this is the Gauss global decomposition of GL nGL_n.

In a similar manner one may discuss subgroups of GL nGL_n by inducing the decomposition from GL nGL_n.

In the case of SL n ( ) SL_n(\mathbb{C}) , and some other cases, the matrices corresponding to UU in the decomposition for w=1w=1 are forming the corresponding (lower) Borel subgroup.

This should be distinguished from the Bruhat decomposition where one wants G=UwAG = U w A instead of G=wUAG = w U A. Except for the case when w=1w=1 when the two decompositions coincide, the matrices which decompose for given ww make a subset of higher codimension, hence nonprincipal (that is w1w\neq 1) Bruhat cells are not open. Furthermore, while in Gauss case the cells make an open cover of GL nGL_n, in Bruhat case they make a partition of GL nGL_n into disjoint subsets of elements.

If BB is a subgroup of lower triangular matrices, then for the fixed ww, the entries of UU as a function of GG in the decomposition g=wUAg=wUA are the coordinates on the patch in GL n/BGL_n/B evaluated at the coset of gg. The decomposition therefore for fixed ww corresponds to the trivialization of the principal BB-bundle GL nGL n/BGL_n\to GL_n/B over the open Gauss cell corresponding to ww.


Named after Carl Friedrich Gauß.

See also:

Last revised on June 18, 2024 at 10:55:35. See the history of this page for a list of all contributions to it.