I have a vector $\boldsymbol{v} \in \mathbb{R}^n$ that is an eigenvector of some matrix $\boldsymbol{A}$ with eigenvalue 1. So $\boldsymbol{v} \in E_1^A$ i.e. eigenspace of A with eigenvalue 1.
Suppose $\boldsymbol{v}$ is a linear combination of $p$ vectors:
$$\boldsymbol{v} = \beta_1 \boldsymbol{v}_1 + \beta_2 \boldsymbol{v}_2 + \cdot\cdot\cdot + \beta_p \boldsymbol{v}_p$$
Are the $\boldsymbol{v}_i \in E_1^A$ as well?
Not necessarily, consider the matrix $$A = \begin{bmatrix}1& 0\\0 & 2\end{bmatrix}$$ which has the eigenvector $\begin{bmatrix}1\\0\end{bmatrix} \in E^A_1$, but $$\begin{bmatrix}1\\0\end{bmatrix} = \begin{bmatrix}0.5\\0.5\end{bmatrix} + \begin{bmatrix}0.5\\-0.5\end{bmatrix}.$$