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

TL;DR

This paper analyzes the global convergence of a specific differential equation on the special orthogonal group, summarizing and extending existing results in a self-contained manner.

Contribution

It provides a comprehensive, self-contained summary and extension of known convergence results for a gradient flow on SO(n).

Findings

The differential equation converges globally on SO(n).

Existing results are unified and extended in a self-contained presentation.

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 $f:SO(n)\rightarrow\mathbb{R},\Theta\mapsto{2n-2\tr{\Theta}}$. 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.