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Preprint 760-2002

A general framework for the perturbation theory of matrix equations

Source file is available as :   Postscript Document

Author(s) : Mihail Konstantinov , Volker Mehrmann , Petko Petkov , Dawei Gu

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 760-2002

MSC 2000

15A24 Matrix equations and identities
93C73 Perturbations

Abstract :
A general framework is presented for the local and non-local perturbation analysis of general real and complex matrix equations in the form $F(P,X) = 0$, where $F$ is a continuous, matrix valued function, $P$ is a collection of matrix parameters and $X$ is the unknown matrix. The local perturbation analysis produces condition numbers and improved first order homogeneous perturbation bounds for the norm $\|\de X\|$ or the absolute value $|\de X|$ of $\de X$. The non-local perturbation analysis is based on the method of Lyapunov majorants and fixed point principles. % for the operator $\pi(p,\cdot)$. It gives rigorous non-local perturbation bounds as well as conditions for solvability of the perturbed equation. The general framework can be applied to various matrix perturbation problems in science and engineering. We illustrate the procedure with several simple examples. Furhermore, as a model problem for the new framework we derive a new perturbation theory for continuous-time algebraic matrix Riccati equations in descriptor form, $Q + A^HXE + E^HXA - E^HXSXE = 0$. The associated equation $Q + A^HXE + E^HX^HA - E^HX^HSXE = 0$ is also briefly considered.

Keywords : Perturbation analysis, general matrix equations,descriptor Riccati equations.

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