Q:

prove $GL(n,R)\cong O(n)\times R_{+}^n\times R^{\frac{n(n-1)}{2}}$

$GL(n,R)$ is the general linear group ,$O(n)$ is the orthogonal group,how to prove $GL(n,R)\cong O(n)\times R_{+}^n\times R^{\frac{n(n-1)}{2}}$

A:

According to the Iwasawa decomposition, any element $T$ of $\mathrm{GL}_n(\mathbb{R})$ can be written in the form $T=KAN$ where $K\in O(n)$, $A$ is a diagonal matrix with positive entries, and $N$ is unipotent (i.e. upper triangular with 1s on the diagonal). Then the space of diagonal matrices with positive entries can be identified with $\mathbb{R}_+^n$, and the unipotent matrices with $\mathbb{R}^{\frac{n(n-1)}{2}}$.

