Lie Groups pp 197-204 | Cite as

# The Iwasawa Decomposition

## Abstract

Let us begin this topic with an example. Let *G =* GL(*n*, ℂ). It is the complexification of *K*= *U*(*n*), which is a maximal compact subgroup. Let *T* be the maximal torus of *K* consisting of diagonal matrices whose eigenvalues have absolute value 1. The complexification *T* _{ℂ} of *T* can be factored as *TA*, where *A* is the group of diagonal matrices whose eigenvalues are positive real numbers. Let *B* be the group of upper triangular matrices in *G*,and let *B* _{0} be the subgroup of elements of *B* whose diagonal entries are positive real numbers. Finally, let *N* be the subgroup of unipotent elements of *B*. Recalling that a matrix is called *unipotent* if its only eigenvalue is 1, the elements of *N* are upper triangular matrices whose diagonal entries are all equal to 1. We may factor *B* = *TN* and *B* _{0} = *AN*. The subgroup *N* is normal in *B* and *B* _{0}, so these decompositions are semidirect products.