Iwasawa decomposition

The Iwasawa decomposition of semisimple Lie groups generalizes the fact that any square matrix can be represented as the product of an orthogonal matrix and an upper triangular matrix in a unique way. It is named after Kenkichi Iwasawa (1949 ), who introduced it for real semisimple Lie groups.

Special case: Matrices

A special case is the unique representation of each element of the particular linear array as a product of three elements.

Be the special orthogonal group, the set of diagonal matrices with positive diagonal entries whose product is 1, and the amount of triangular matrices, whose diagonal ones are everywhere. Then there exist uniquely determined for each, and such that.

General case

Let be a semisimple Lie group. Then there is a decomposition

With a compact subgroup of an abelian subgroup and a nilpotent subgroup, so that each element in a unique way as a product

Can decompose with.

The decomposition is not uniquely determined. Each separation with the above properties is Iwasawa decomposition.

The method is named after its developer Iwasawa Kenkichi.