On the differential equation $\dot{\Theta }=({\Theta }^T-\Theta )\Theta$ with $\Theta \in SO(n)$

Abstract

In this note we consider the global convergence properties of the differential equation $\dot{\Theta }=({\Theta }^T-\Theta )\Theta$ with $\Theta \in SO(n)$, which is a gradient flow of the function $f:SO(n)\rightarrow\mathbb{R},Θ\mapsto{2n-2\trΘ}$. Many of the presented results are not new, but scattered throughout literature. The motivation of this note is to summarize and extend the convergence results known from literature. Rather than giving an exhaustive list of references, the results are presented in a self-contained fashion.