Bounds for the variation of matrix eigenvalues and polynomial roots

Abstract

For two given complex matrices A, B, upper bounds are derived for the optimal matching distance between the spectra σ( A) and σ( B) in terms of ‖ A - B 2‖, where ‖ ⋅ ‖ 2 is the spectral norm. The case of arbitrary matrix norms is treated. A similar result estimates the optimal matching distance between the roots of two polynomials. These bounds replace a factor of 4 in earlier results by the value 16 (3√3) ≈3.08 .