  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.
