In many applications, we encounter the problem of reducing
a large matrix to a much smaller one.
Theoretically the answer to such a question  is quite simple
and can be found through singular value decomposition (SVD).
However, the SVD algorithms, both the traditional ones based on
diagonalization and the recently developed  randomized algorithms require
cubic scaling in terms of computational complexity.
In this talk, we will focus on the case when the matrix comes
from discretization of differential operators or the generator
of local Markov chains on large networks, and we will
present linear scaling algorithms.
Key to such linear scaling algorithms is the fact that there exist
very localized bases for the eigen-space of interest.
We will also discuss application to electronic structure analysis.