On the hyper-lyapunov inclusions

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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.
Original languageEnglish
Number of pages19
StateSubmitted - 21 Apr 2021


  • math.FA
  • math.OC
  • 34H05 47N70 93B20 93C15


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