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 ?



Generalized Rayleigh-quotient gradient flow on Grassmannian
On the Stiefel manifold, consider the function $f(X)=\operatorname{tr}(X^TAX)$. It evolves under the flow of the given vector field as
$$\frac d{dt}f=2\operatorname{tr}(X^TA(1-XX^T)AX)\;.$$
Because $1-XX^T$ describes the projection onto the orthogonal complement of the span $V\_X$ of the columns of $X$, we have
\begin{gather\*}\frac d{dt}f\ge 0\;,\\\frac d{dt}f=0\iff \dot X=0\iff\text{$V\_X$ is $A$-invariant.}\end{gather\*}
Hence, $X$ is a generalised gradient field for $f$ (it might in fact be half the actual gradient, but I have not checked).


Now it is easy to see that $f$ induces a Morse-Bott function on the Grassmannian (Morse if the eigenvalues of $A$ are distinct). The only stable fixpoints therefore correspond to the maximum, and the points that don't flow into one of the maxima form a stratified subset of lower dimension.


