I am working on a product in Morse-Bott homology which has led me to the following considerations and unanswered question. I would be very grateful if anyone could help.

Suppose $H:\mathbb{R}^n \to \mathbb{R}^n$ is a linear map which i symmetric when viewed as a matrix. The spectral theorem gives a decomposition $\mathbb{R}^n\cong E^+ \oplus E^- \oplus E^0$, where $E^{\pm}$ denotes the sum of positive (+) respectively negative $(-)$ eigenspaces of $H$ and $E^0$ denotes the kernel of $H$.

Let $\mathcal{M}$ denote the set of symmetric, positive definite matrices on $\mathbb{R}^n$. I am interested in the $n\times n$ matrix $g^{-1}H$ for $g\in \mathcal{M}$ (corresponding to the Hessian in Morse theory). Denote by $X_g:\mathbb{R}\to \mathbb{R}^{n\times n}$ the path of matrices \begin{align} \mathbb{R}&\to \mathbb{R}^{n\times n} \\ t &\mapsto e^{g^{-1}Ht}. \end{align}

Now define the stable subspace $V^s(g)$ of the operator $X_g$ by \begin{align} V^s(g):=\{v\in \mathbb{R}^n \ |\ X_g(t)v \to 0 \ \text{for} \ t\to \infty \}. \end{align} My question is: How big is the set $\mathcal{S}:=\{ V^s(g)\ |\ g\in \mathcal{M}\}$ in the Grassmannian of $\mathbb{R}^n$?

For example, is it true that any subspace of maximal dimension sitting inside the cone generated by $E^+$ in $\mathbb{R}^n\cong E^+ \oplus E^- \oplus E^0$ is an element of $\mathcal{S}$?

Any help is much appreciated!!! Thanks in advance,

Mads