Example 7

Iwasawa decomposition of $\mathbf{GL}(n, K)$.

In this example, K denotes one of the fields $\mathbf{R}, \mathbf{C}, \mathbf{H}$. If $\lambda \in K$, $\overline{\lambda}$ is defined to be equal to $\lambda$ if $K = \mathbf{R}$, and to the conjugate of $\lambda$ if $K = \mathbf{C}$ or $\mathbf{H}$. Let E be a right vector space over K of dimension $n$, and let $\Phi$ be a nondegenerate positive hermitian form on E.