TITLE: Compute only selected components of an eigenvector
QUESTION [7 upvotes]: 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?

REPLY [4 votes]: Here is  some information that you might find useful. The context is that of PageRank computation, for the purpose of updating only some of ranks (essentially entries of the eigenvector).

Y. Y. Chen, 2004. Local methods for estimating PageRank values
Fast incremental and personalized page rank

In general, chasing references for "incremental" pagerank or just "top few ranks" should help you find more related work and ideas.
Alternatively, you could try to solve the problem approximately by using randomization to subsample the matrix $A$, and then using perturbation analysis to estimate the error of approximation --- some ideas for this can be found by following work related to the so-called "CUR" decomposition.