Fabian Wirth
The generalized spectral radius is strictly increasing.
Preprint series:
MSC:
15A99, 34D08, 93D09
Abstract:
Using a result linking convexity and irreducibility of matrix sets it is shown that the generalized spectral radius of a compact set of matrices is a strictly increasing function of the set in a very natural sense. As an application some consequences of this property in the area of time-varying stability radii are discussed. In particular, using the implicit function theorem sufficient conditions for Lipschitz continuity are derived. An example is presented of a linearly increasing family of matrix polytopes for which the proximal subgradient of the generalized spectral radius at a certain polytope contains $0$, so that the implicit function theorem is not applicable in all cases.

Keywords: Generalized spectral radius, joint spectral radius, irreducibility, monotonicity, stability radius.

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Home Fabian Wirth's publication page Research Group "Regelungssysteme"
Zentrum für Technomathematik (German)
Institute for Dynamical Systems
The Maths Department
The University of Bremen

Notes:
accepted for publication by Linear Algebra and its Applications