  We prove a new theorem relating the number of distinct eigenvalues of a
matrix after perturbation to the prior number of distinct eigenvalues, the rank
of the update, and the degree of nondiagonalizability of the matrix. In
particular, a rank one update applied to a diagonalizable matrix can at most
double the number of distinct eigenvalues. The theorem applies to both
symmetric and nonsymmetric matrices and perturbations, of arbitrary magnitudes.
An an application, we prove that in exact arithmetic the number of Krylov
iterations required to exactly solve a linear system involving a diagonalizable
matrix can at most double after a rank one update.
