A new Gershgorin-type result for the localisation of the spectrum of matrices

We present a Gershgorin's type result on the localisation of the spectrum of a matrix. Our method is elementary and relies upon the method of Schur complements, furthermore it outperforms the one based on the Cassini ovals of Ostrovski and Brauer. Furthermore, it yields estimates that hold without major differences in the cases of both scalar and operator matrices. Several refinements of known results are obtained.