Let $H$ be the set of real symmetric $n\times n $ matrices, and $\mathbb{P}=\{x\in H|\,x^2=x,\,tr(x)=1\}$, $\mathcal{P}=\{x\in H|\,x^2=(tr(x))x,\,tr(x)>1\}$.
You also have some information: $\mathbb{P}$ is isomorphic to $\mathbb{R}P^{n-1}$, then $\mathbb{P}$ is a compact manifold of dimension $(n-1)$, also $\mathcal{P}$ is diffeomorphic to $(0,\infty)\times \mathbb{P}$.
Then for any $x\in H$, we let $L_x$ be the endomorphism on $H$ which maps $u$ to $xu:= \dfrac{u\cdot x+x\cdot u}{2}$, "$\cdot$" is the ordinary matrix product. Show that $T_x\mathcal{P}=\{x\}\times image(L_x)$, for any $x\in\mathcal{P}$.
(I'm thinking a local parametrization and compute the image of its differential, which is the definition of tangent space, but still do not have much information.)