I am wondering whether it is possible to compute portions of the eigenvectors of a given (possibly very big) matrix. More formally, consider the eigenvalue problem $\mathbf{Ax} = \lambda \mathbf{x}$, where $\mathbf{A}$ is $n \times n$ Hermitian. For a fixed eigenvector $\mathbf{x}$, I am only interested in the values $\mathbf{x}_k$ for some choices of $k \in \{1,\dots,n\}$.

Is it possible to restrict the computation as above? If not, is it possible to obtain an approximate solution, and under which conditions?