ELA, Volume 15, pp. 239-250, August 2006, abstract.

An Eigenvalue Inequality and Spectrum Localization 
for Complex Matrices

Maria Adam and Michael J. Tsatsomeros

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.