An optimal bound for the spectral variation of two matrices

AbstractWe establish a bound for the spectral variation of two complex n × n matrices A,B in terms of ∥A∥, ∥B∥, and ∥A − B∥. Here ∥ ∥ denotes the spectral norm. It is always better than a bound previously given by Bhatia and Friedland, and it is optimal. We describe the set of pairs A,B for which the bound is attained