An eigenvalue inequality and spectrum localization for complex matrices

Abstract

Using the notions of the numerical range, Schur complement and unitary equivalence,
an eigenvalue inequality is obtained for a general complex matrix, giving rise to a region in the
complex plane that contains its spectrum. This region is determined by a curve, generalizing and
improving classical eigenvalue bounds obtained by the Hermitian and skew-Hermitian parts, as well as the numerical range of a matrix.