The following theorem appears without proof in :

Helmke, Uwe, and John B. Moore. Optimization and dynamical systems. Springer Science & Business Media, 2012.

Let $A$ be a symmetric $n\times n$ real matrix. Define the Stiefel manifold as $St(k,n)=\{X\in \mathbb{R}^{n\times k}|X^TX=I\}$.

Then, we consider the following equation :

$\dot{X}=(I-XX^T)AX$, where $X\in St(k,n)$

It is a matrix ODE which is invariant under right multiplication by $O(k)$. Hence it can be considered as an ODE on the Grassmannian.

Helmke and Moore state that it almost-surely converges to an $A$-invariant subspace spanned by a dominant $k$-dimensional eigenbasis of $A$.

I am looking for proof of this theorem. Does anyone know any suitable references ?