Orthogonal matrices form a compact set [duplicate]

Prove that the set of all $n \times n$ orthogonal matrices is a compact subset of $\mathbb{R}^{n^2}$.

I don't know how it can be done. Thanks.

Solution 1:

Let $M^{n\times n}$ be the set of all matrices and $\mathcal{O}$ the subset of orthogonal matrices. Define $f\colon M\to M,\;A\mapsto A^\intercal A$. Then $\mathcal{O}=f^{-1}(I)$, where $I$ is the identity matrix. Since $f$ is continuous, $\mathcal{O}$ is closed as a preimage of a singleton. Since $\|Qx\|=\|x\|$ for each orthogonal matrix $Q$, $\mathcal{O}$ lies in the 1-ball, i.e. is bounded. Heine-Borel says that $\mathcal{O}$ is compact. Since all norms in finite-dimensional vectorspaces are equivalent it doesn't matter that we used the operatornorm