  An upper bound for the number of distinct eigenvalues of a perturbed matrix
has been recently established by P. E. Farrell [1, Theorem 1.3]. The estimate
is the central result in Farrell's work and can be applied to estimate the
number of Krylov iterations required for solving a perturbed linear system. In
this paper, we present an improved upper bound for the number of distinct
eigenvalues of a matrix after perturbation. Furthermore, some results based on
the improved estimate are presented.
