On the hyper-lyapunov inclusions

TL;DR

This paper refines the Gantmacher-Lyapunov theorem by introducing Hyper-Lyapunov inclusions, characterizing matrices with spectra in specific disks within the right-half plane, and explores their implications for matrix computations.

Contribution

It introduces Hyper-Lyapunov inclusions formulated through quadratic matrix inequalities, extending classical spectral characterizations and linking them to matrix sign function iterations.

Findings

Disks are closed under inversion, generalizing spectral regions.

Hyper-Lyapunov inclusions recover classical results as radius approaches infinity.

Application to matrix sign function iteration schemes.

Abstract

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.