nLab Gauss decomposition

Idea

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

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

1. $L$ is a lower triangular unidiagonal matrix,

2. $U$ is upper triangular unidiagonal

3. $D$ is diagonal.

Here unidiagonal means with only unit entries on the diagonal.

(If $D$ 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.

Properties

Existence

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 $U$ or $L$ involves ratios of (determinants of) minors of the same size.

For invertible matrices

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

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

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

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

In a similar manner one may discuss subgroups of $GL_n$ by inducing the decomposition from $GL_n$.

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

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

If $B$ is a subgroup of lower triangular matrices, then for the fixed $w$, the entries of $U$ as a function of $G$ in the decomposition $g=wUA$ are the coordinates on the patch in $GL_n/B$ evaluated at the coset of $g$. The decomposition therefore for fixed $w$ corresponds to the trivialization of the principal $B$-bundle $GL_n\to GL_n/B$ over the open Gauss cell corresponding to $w$.

Related concepts

• matrix decomposition

  • QR decomposition

  • LU decomposition

  • polar decomposition

• quantum Gauss decomposition.

Literature

Named after Carl Friedrich Gauß.

See also:

• eom, Gauss decomposition