Distance to $O_n$ and polar decomposition (statement)

Polar decomposition. Every real $A$ is $U\Sigma V^T$ with $U,V$ orthogonal and $\Sigma=\operatorname{diag}(\sigma_1,\ldots,\sigma_n)\ge0$.

Lower bound. For any $Q\in O_n$, $\|A-Q\|_2=\|\Sigma-V^TQU\|_2=\|\Sigma-R\|_2$ where $R=V^TQU\in O_n$. Since $R$ is orthogonal, its singular values are all $1$, so $\|\Sigma-R\|_2\ge\max_i|\sigma_i-1|$.

Upper bound. Setting $Q=UV^T\in O_n$, one has $\|A-Q\|_2=\|\Sigma-I\|_2=\max_i|\sigma_i-1|$.

Conclusion. $\operatorname{dist}_2(A,O_n)=\max_i|\sigma_i(A)-1|$.