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



Transversality in Morse theory, linear algebra version
Consider $H$, $g$ as symmetric bilinear forms, not as endomorphisms.
Then $g^{-1}H$ is an endomorphism that describes the gradient vector field.
Note that $\ker H=\ker g^{-1}H$ is a well-defined subspace.
Moreover, the dimensions $n\_\pm$ of the positive/negative eigenspaces are determined by $H$ alone.


Every pair of subspaces $V^-$, $V^+$ of dimension $n\_-$, $n\_+$ such that $\pm H|\_{V^\pm}>0$ simultaneously occur as stable and unstable subspace $V^s(g)$, $V^u(g)$ for some $g\in\mathcal M$: Define $g$ such that these spaces are orthogonal to each other and to $\ker H$. On $V^\pm$, put $g=\pm H^{-1}$, on $\ker H$ choose any metric you like.


