  This paper is about the relation of random matrix theory and the
subordination phenomenon in complex analysis. We find that the resolvent of the
sum of two random matrices is approximately subordinated to the resolvents of
the original matrices. We estimate the error terms in this relation and in the
subordination relation for the traces of the resolvents. This allows us to
prove a local limit law for eigenvalues and a delocalization result for
eigenvectors of the sum of two random matrices. In addition, we use
subordination to determine the limit of the largest eigenvalue for the rank-one
deformations of unitary-invariant random matrices.
