FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA
(FUNDAMENTAL AND APPLIED MATHEMATICS)
1999, VOLUME 5, NUMBER 4, PAGES 1111-1133
A. A. Mailybaev
Abstract
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Families of matrices, smoothly dependent on a vector of parameters,
are considered.
V. I. Arnold (1971) has found normal forms of families
of complex matrices (miniversal
deformations), such that any family of matrices
in the vicinity of a point
can be transformed to them
by smoothly dependent on the vector of parameters
change of basis and smooth change of parameters. Miniversal deformations of
real matrices have been studied by D. M. Galin (1972).
In this paper a method of determining functions describing
change of basis and change of parameters, transforming arbitrary family
to the miniversal deformation, is suggested. The functions are
found as Taylor
series, where derivatives of the functions are determined from
a recurrent procedure
using derivatives of these functions of lower orders and derivatives of
the family. Examples are given.
The results obtained allow to use miniversal deformations
for investigation of
different properties of matrix families more efficiently. This is
shown in the paper, where tangent cones to the stability domain
(linear approximations) at boundary points are found.
All articles are published in Russian.
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Last modified: December 9, 1999