Inhalt des Dokuments
Preprint 760-2002
A general framework for the perturbation theory of matrix equations
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.