  The eigenproblem of low-rank updated matrices are of crucial importance in
many applications. Recently, an upper bound on the number of distinct
eigenvalues of a perturbed matrix was established. The result can be applied to
estimate the number of Krylov iterations required for solving a perturbed
linear system. In this paper, we revisit this problem and establish some
refined bounds. Some {\it a prior} upper bounds that only rely on the
information of the matrix in question and the low-rank update are provided.
Examples show the superiority of our theoretical results over the existing
ones. The number of distinct singular values of a matrix after perturbation is
also investigated.
