Gauss decomposition

Let kk be a field and gM n(k)g\in M_n(k) n×nn\times n square matrix whose entries are in kk. One would like to decompose this matrix as g=UDLg = UDL where LL is the lower triangular unidiagonal matrix, UU upper triangular unidiagonal and DD a diagonal matrix. Unidiagonal means with only units on a diagonal. One can also try to decompose g=UAg = UA where AA is lower triangular and UU upper triangular unidiagonal. Not every matrix however has such a decomposition: one tries to find the solution via 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, while the solution for AA the ratio of (determinants of) minors of sizes different by one. These decompositions of a matrix GG are called the Gauss decompositions.

If gGL n(k)g\in GL_n(k) then there exist a permutation n×nn\times n matrix ww such that the problem G=wUDLG = wUDL has a solution. 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), one of the “Gauss cells”; the decomposition of matrices is for each ww also called Gauss decomposition. Then w=1w=1 is said to be the main cell. The group GL n(k)GL_n(k) is covered by n!n! open subsets, this is the Gauss of global decomposition of GL nGL_n; similarly one can do for various 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 AA in the decomposition for w=1w=1 are forming the corresponding (lower) Borel subgroup.

This should be distinguished from Bruhat decomposition where one wants g=UwAg=UwA instead of g=wUAg=wUA. 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 a 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.

Gauss decomposition can be generalized to the matrices with noncommutative entries and there are explicit formulas involving quasideterminants, and in the case of quantum linear groups in terms of quantum minors.

Remark. The upper conventions are more from mathematical physics literature. The lowest weight representations have often positive energy, what is normal requirement there and the lower triangular matrices are used to act on the lowest weight vector. In mathematical community, one prefers to talk about upper triangular matrices forming Borel subgroup and about the decomposition LDULDU and LALA where AA is upper triangular and act on heighest weight representations instead.

See also quantum Gauss decomposition.

Last revised on December 16, 2009 at 19:43:40. See the history of this page for a list of all contributions to it.