On the variation of the spectrum of a normal matrix Original Research Article

Let A and à be two n × n normal matrices with spectra {λ} and {image}. Then by the Hoffman-Wielandt theorem, there is a permutation π of {1,2,…,n} such that image, where F denotes the Frobenius norm. However, if A is normal but ˜A nonnormal, it may be asked: How to relate the eigenvalues of ˜A to those of A? An answer is given in this paper: There is a permutation π of {1,2,…,n} such that image and the factor √n is best possible. As a corollary, we have imageλreverse similarφ(j) − λj less-than-or-equals, slant n short parallel Areverse similar − Ashort parallel2, for the spectral norm 2. Thus, the known upper bound (2n − 1)˜A − A2 is reduced by a factor of about two.