On the variation of the spectrum of a normal matrix

AbstractLet A and à be two n × n normal matrices with spectra {λ} and {λj}. Then by the Hoffman-Wielandt theorem, there is a permutation π of {1,2,…,n} such that∑ nj=1 |λ∽φ(j) − λj|2 ⩽ ∥ A∽ − A∥F,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∑ nj=1 |λ∽φ(j) − λj|2 ⩽ n∥ A∽ − A∥Fand the factor √n is best possible. As a corollary, we have|λ∽φ(j) − λj| ⩽ n ∥ A∽ − A∥2,for the spectral norm | |2. Thus, the known upper bound (2n − 1)|~A − A|2 is reduced by a factor of about two