• The
Jordan–Chevalley decomposition of an element in algebraic group as a product of semisimple and unipotent elements • The
Bruhat decomposition G=BWB of a
semisimple algebraic group into double
cosets of a
Borel subgroup can be regarded as a generalization of the principle of
Gauss–Jordan elimination, which generically writes a matrix as the product of an upper triangular matrix with a lower triangular matrix—but with exceptional cases. It is related to the Schubert cell decomposition of
Grassmannians: see
Weyl group for more details. • The
Cartan decomposition writes a semisimple real Lie algebra as the sum of eigenspaces of a
Cartan involution. • The
Iwasawa decomposition G=KAN of a semisimple group G as the product of
compact, abelian, and
nilpotent subgroups generalises the way a square real matrix can be written as a product of an
orthogonal matrix and an
upper triangular matrix (a consequence of
Gram–Schmidt orthogonalization). • The
Langlands decomposition P=MAN writes a parabolic subgroup P of a Lie group as the product of semisimple, abelian, and nilpotent subgroups. • The
Levi decomposition writes a finite dimensional Lie algebra as a
semidirect product of a
solvable ideal and a
semisimple subalgebra. • The
LU decomposition of a dense subset in the general linear group. It can be considered as a special case of the
Bruhat decomposition. • The
Birkhoff decomposition, a special case of the
Bruhat decomposition for affine groups. ==References==