Partial Indices of Small Perturbations of a Degenerate Continuous Matrix Function

2002 ◽  
pp. 185-195
Author(s):  
Israel Feldman ◽  
Naum Krupnik ◽  
Alexander Markus
Keyword(s):  
Matrix Function ◽  
Small Perturbations ◽  
Partial Indices

scholarly journals On explicit Wiener–Hopf factorization of 2 × 2 matrices in a vicinity of a given matrix

2020 ◽  
Vol 476 (2238) ◽  
pp. 20200027 ◽  
Author(s):  
L. Ephremidze ◽  
I. Spitkovsky
Keyword(s):  
Numerical Simulations ◽  
Matrix Function ◽  
Factorization Method ◽  
Spectral Factorization ◽  
Engineering Applications ◽  
Partial Indices ◽  
Matrix Spectral Factorization

As it is known, the existence of the Wiener–Hopf factorization for a given matrix is a well-studied problem. Severe difficulties arise, however, when one needs to compute the factors approximately and obtain the partial indices. This problem is very important in various engineering applications and, therefore, remains to be subject of intensive investigations. In the present paper, we approximate a given matrix function and then explicitly factorize the approximation regardless of whether it has stable partial indices. For this reason, a technique developed in the Janashia–Lagvilava matrix spectral factorization method is applied. Numerical simulations illustrate our ideas in simple situations that demonstrate the potential of the method.

scholarly journals Exact conditions for preservation of the partial indices of a perturbed triangular 2 × 2 matrix function

2020 ◽  
Vol 476 (2237) ◽  
pp. 20200099 ◽  
Author(s):  
Victor M. Adukov ◽  
Gennady Mishuris ◽  
Sergei V. Rogosin
Keyword(s):  
Unit Circle ◽  
Matrix Function ◽  
Triangular Matrix ◽  
Specific Class ◽  
Matrix Functions ◽  
Approximate Methods ◽  
Parametric Space ◽  
Partial Indices ◽  
Unstable Set ◽  
Factorization Technique

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.

scholarly journals On effective criterion of stability of partial indices for matrix polynomials

2020 ◽  
Vol 476 (2238) ◽  
pp. 20200012
Author(s):  
N. V. Adukova ◽  
V. M. Adukov
Keyword(s):  
Small Perturbation ◽  
Stability Criterion ◽  
Matrix Function ◽  
Toeplitz Matrices ◽  
Matrix Functions ◽  
Matrix Polynomials ◽  
Partial Indices ◽  
The Stability ◽  
The Given

In the work, we obtain an effective criterion of the stability of the partial indices for matrix polynomials under an arbitrary sufficiently small perturbation. Verification of the stability is reduced to calculation of the ranks for two explicitly defined Toeplitz matrices. Furthermore, we define a notion of the stability of the partial indices in the given class of matrix functions. This means that we will consider an allowable small perturbation such that a perturbed matrix function belong to the same class as the original one. We prove that in the class of matrix polynomials the Gohberg–Krein–Bojarsky criterion is preserved, i.e. new stability cases do not arise. Our proof of the stability criterion in this class does not use the Gohberg–Krein–Bojarsky theorem.

scholarly journals Regular approximate factorization of a class of matrix-function with an unstable set of partial indices

2018 ◽  
Vol 474 (2209) ◽  
pp. 20170279 ◽  
Author(s):  
G. Mishuris ◽  
S. Rogosin
Keyword(s):  
Matrix Function ◽  
Stable Set ◽  
Matrix Functions ◽  
Stable Sets ◽  
Approximate Factorization ◽  
Original Matrix ◽  
Partial Indices ◽  
Unstable Set ◽  
The Matrix ◽  
The Difference

From the classic work of Gohberg & Krein (1958 Uspekhi Mat. Nauk. XIII , 3–72. (Russian).), it is well known that the set of partial indices of a non-singular matrix function may change depending on the properties of the original matrix. More precisely, it was shown that if the difference between the largest and the smallest partial indices is larger than unity then, in any neighbourhood of the original matrix function, there exists another matrix function possessing a different set of partial indices. As a result, the factorization of matrix functions, being an extremely difficult process itself even in the case of the canonical factorization, remains unresolvable or even questionable in the case of a non-stable set of partial indices. Such a situation, in turn, has became an unavoidable obstacle to the application of the factorization technique. This paper sets out to answer a less ambitious question than that of effective factorizing matrix functions with non-stable sets of partial indices, and instead focuses on determining the conditions which, when having known factorization of the limiting matrix function, allow to construct another family of matrix functions with the same origin that preserves the non-stable partial indices and is close to the original set of the matrix functions.

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