In mathematics, the Iwasawa decomposition (aka KAN from its expression) of a semisimple Lie group generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix (QR decomposition, a consequence of GramâÂÂSchmidt orthogonalization). It is named after Kenkichi Iwasawa, the Japanese mathematician who developed this method.
Then the Iwasawa decomposition of is
and the Iwasawa decomposition of G is
meaning there is an analytic diffeomorphism (but not a group homomorphism) from the manifold to the Lie group , sending .
The dimension of A (or equivalently of ) is equal to the real rank of G.
Iwasawa decompositions also hold for some disconnected semisimple groups G, where K becomes a (disconnected) maximal compact subgroup provided the center of G is finite.
The restricted root space decomposition is
where is the centralizer of in and is the root space. The number is called the multiplicity of .
If G=SL<sub>n</sub>(R), then we can take K to be the orthogonal matrices, A to be the positive diagonal matrices with determinant 1, and N to be the unipotent group consisting of upper triangular matrices with 1s on the diagonal.
For the case of n = 2, the Iwasawa decomposition of G = SL(2, R) is in terms of
For the symplectic group G = Sp(2n, R), a possible Iwasawa decomposition is in terms of
Obtaining the matrices appearing in the decomposition above can be reduced to the calculation of matrix square roots, matrix inverses and performing a QR decomposition.
There is an analog to the above Iwasawa decomposition for a non-Archimedean field : In this case, the group can be written as a product of the subgroup of upper-triangular matrices and the (maximal compact) subgroup , where is the ring of integers of .