Consider the manifold $\mathbb{CP}^{2}$. Let $l \in \mathbb{CP}^{2}$ be arbitrary. Define $F: \mathbb{C}^{3} \rightarrow \mathbb{C}^{3}$ to be the reflection in the line $l \subset \mathbb{C}^{3}$.
I have now found the following claim:
The tangent space $T_{l}\mathbb{CP}^{2}$ can be identified with the space of Hermitian operators $A$ on $\mathbb{C}^{3}$ such that $AF + FA = 0$.
It is unclear to me how this identification would go. Let's choose a unit vector $z \in l$. My guess would be that we may identify a smooth function $f: \mathbb{CP}^{2} \rightarrow \mathbb{R}$ with a unique smooth function $\tilde{f}: \mathbb{C}^{3}\setminus \{ 0 \} \rightarrow \mathbb{R}$, we could then define the derivation \begin{equation} A \triangleright f = \frac{\text{d}}{\text{d}t}\bigg|_{t=0} \tilde{f}\left(z + t A z \right). \end{equation}
Question: Is this correct? If no, how does the identification work? If yes, how to show this...