Abstract
The possible instability of partial indices is one of the important constraints in the creation of approximate methods for the factorization of matrix functions. This paper is devoted to a study of a specific class of triangular matrix functions given on the unit circle with a stable and unstable set of partial indices. Exact conditions are derived that guarantee a preservation of the unstable set of partial indices during a perturbation of a matrix within the class. Thus, even in this probably simplest of cases, when the factorization technique is well developed, the structure of the parametric space (guiding the types of matrix perturbations) is non-trivial.
Highlights
The factorization problem involves the representation of a square nonsingular matrix function G ∈ G(M(Γ ))n×n, defined on a
Non-singular matrices G−(t), G+(t) possess, together with their inverses, analytic continuations into D− and D+, respectively, where D−, D+ are the domains on the Riemann sphere lying, respectively, to the right and to the left of the curve Γ, with reference to the orientation chosen for Γ
In our paper we have discussed the behaviour of the partial indices of a given matrix function A(t) under perturbations from the ε-neighbourhood in different classes of matrix functions
Summary
The (right) factorization problem involves the representation of a square nonsingular matrix function G ∈ G(M(Γ ))n×n, defined on a. An essential constraint in this respect was found independently by Gohberg & Krein [10] and by Bojarski [11] They introduced the notion of a stable set of partial indices for a non-singular matrix function (those that preserve their values with a small perturbation of the matrix). The study of this problem was initiated by Bojarski [11] He demonstrated that the set of matrices with the same partial indices is in general not an open set in the class of invertible matrix functions with a given order of smoothness, but is still a connected set. Rigorous proofs for some of the statements are given in appendix A
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