TY - UNPB
T1 - On the hyper-lyapunov inclusions
AU - Lewkowicz, Izchak
N1 - 19 pages
PY - 2021/4/21
Y1 - 2021/4/21
N2 - 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.
AB - 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.
KW - math.FA
KW - math.OC
KW - 34H05 47N70 93B20 93C15
U2 - https://doi.org/10.48550/arXiv.2009.13283
DO - https://doi.org/10.48550/arXiv.2009.13283
M3 - Preprint
SP - 1
EP - 19
BT - On the hyper-lyapunov inclusions
ER -