TITLE: Generalized Rayleigh-quotient gradient flow on Grassmannian
QUESTION [7 upvotes]: 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 ?

REPLY [5 votes]: 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.