TITLE: Prove that $O(n)$ is a maximal compact subgroup of $GL(n,\mathbb R)$
QUESTION [5 upvotes]: The indication is: Let $P$ is a symmetric positive definite matrix such that the norm of $p^k$ is smaller than a constant $C$ for every integer $k$, then $P=I_n$.

REPLY [9 votes]: Let $K$ be a compact subgroup of $GL(n,\mathbb R)$. Let $\beta:\mathbb R^n\times\mathbb R^n\to\mathbb R$ be any inner product and consider the function $\bar \beta:\mathbb R^n\times\mathbb R^n\to\mathbb R$ such that $\bar \beta(x,y)=\int \beta(kx,ky)\,\mathrm dk$, where $\mathrm dk$ is the normalized Haar measure on $K$ (this makes sense because the function $k\in K\mapsto \beta(kx,ky)$ is continuous. One can easily check that $\bar\beta$ is an inner product on $\mathbb R^n$ and that the action of $K$ on $\mathbb R^n$ preserves $\bar\beta$. This means that $K$ is conjugated to a subgroup of $O(n,\mathbb R)$.