Departure from Normality and Eigenvalue Perturbation Bounds

Perturbation bounds for eigenvalues of diagonalizable matrices are derived that do not depend on any quantities associated with the perturbed matrix; in particular the perturbed matrix can be defective. Furthermore, Gerschgorin-like inclusion regions in the Frobenius are derived, as well as bounds on the departure from normality. 1. Introduction. The results in this paper are based on two eigenvalue bounds for normal and Hermitian matrices by Sun and Kahan, respectively. Sun [10] presents a bound for the eigenvalues of an arbitrarily perturbed normal matrix that does not contain the conditioning of the perturbed eigenvectors. Kahan [7] presents a similar, but better bound for Hermitian matrices. From the ideas in the proofs of these two bounds we derive several results: §2: Bounds on the departure from normality of perturbed normal and Hermitian matrices. §3: Perturbation bounds for eigenvalues of normal matrices whose eigenvalues lie on a line in the complex plane. The bounds are in the Frobenius norm and do not depend on the conditioning of perturbed quantities. §5: Perturbation bounds (in the Frobenius and two-norm) for eigenvalues of diago-nalizable matrices. The bounds do not place any restriction on the perturbed matrix and allow it, for instance, to be defective. They also do not depend on the conditioning of the perturbed eigenvectors, which is advantageous if these are ill-conditioned with respect to inversion. An example in §4 illustrates this. §6: Gerschgorin-like inclusion regions in the Frobenius norm for eigenvalues of complex square matrices. §7: Frobenius norm bounds for the departure of a matrix from normality in terms of the diagonal and off-diagonal elements. For real matrices, we bound the departure from normality by the norm of the off-diagonal elements. Our bounds are simple and cheaper to determine than many existing bounds, cf. [8, 9] and the references in there. §8: A lower bound on eigenvalue perturbations for complex square matrices and for Hermitian matrices.