A perturbation bound for the eigenvalues of a singular diagonalizable matrix

Let A be a singular, diagonalizable matrix with group inverse A#, and let A + E be a perturbation of A. We show that each eigenvalue μ of A + E is an O( A#E 2) relative perturbation of a nonzero eigenvalue of A, unless it is small enough in magnitude to be treated as an O( E 2) perturbation of the zero eigenvalue of A.