Counterexample to a conjecture of Elsner on the spectral variation of matrices

AbstractIn 1985, Elsner proved that the Hausdorff distance Δ between the spectra of two n×n matrices A and B satisfiesΔσ(A),σ(B)n⩽∥A∥+∥B∥n−1∥A−B∥,where ∥·∥ denotes the operator norm with respect to the Euclidean norm on Cn. He further conjectured that the same inequality holds for all operator norms. We disprove this conjecture, and also the weaker conjecture where (∥A∥+∥B∥) is replaced by 2max(∥A∥,∥B∥)