On the hyper-lyapunov inclusions

Gantmacher-Lyapunov Theorem (1950's) characterizes matrices whose spectrum lies in the right-half of the complex plane. Here this result is refined to Hyper-Lyapunov inclusion for matrices whose spectrum lies in some disks within the right-half plane. These disks turn to be closed under inversion, and when their radius approaches infinity, the original result is recovered. Hyper-Lyapunov inclusions are formulated through Quadratic Matrix Inequalities and so are the analogous Hyper-Stein sets of matrices whose spectrum lies within a sub-unit disk. As a by-product, it is shown that these disks closed under inversion, are a natural tool to understanding the Matrix Sign Function iteration scheme, used in matrix computations.