New perturbation bounds for the spectrum of a normal matrix

Let A is an element of C-nxn and (A) over tilde is an element of C-nxn be two normal matrices with spectra {lambda(i)}(i=1)(n) and {(lambda) over tilde (i)}(i=1)(n) respectively. The celebrated Hoffman-Wielandt theorem states that there exists a permutation pi of {1, ... , n} such that (Sigma(n)(i=1) vertical bar(lambda) over tilde (pi(i)) - lambda(i)vertical bar(2))(1/2) is no larger than the Frobenius norm of (A) over tilde - A. However, if either A or (A) over tilde is non-normal, this result does not hold in general. In this paper, we present several novel upper bounds for (Sigma(n)(i=1) vertical bar(lambda) over tilde (pi(i)) - lambda(i)vertical bar(2))(1/2), provided that A is normal and (A) over tilde is arbitrary. Some of these estimates involving the "departure from normality" of (A) over tilde have generalized the Hoffman-Wielandt theorem. Furthermore, we give new perturbation bounds for the spectrum of a Hermitian matrix. (c) 2017 Elsevier Inc. All rights reserved.