Factorization of Continuous Self-Adjoint Matrix Functions on the Unit Circle

1981 ◽  
pp. 113-121
Author(s):  
Kevin F. Clancey ◽  
Israel Gohberg
Keyword(s):  
Unit Circle ◽  
Matrix Functions ◽  
Adjoint Matrix

Factorizations of and extensions to J-unitary rational matrix functions on the unit circle

1992 ◽  
Vol 15 (2) ◽  
pp. 262-300 ◽  
Author(s):  
I. Gohberg ◽  
M. A. Kaashoek ◽  
A. C. M. Ran
Keyword(s):  
Unit Circle ◽  
Matrix Functions ◽  
Rational Matrix

Minimal Pencil Realizations of Rational Matrix Functions with Symmetries

1998 ◽  
Vol 41 (2) ◽  
pp. 178-186 ◽  
Author(s):  
Ilya Krupnik ◽  
Peter Lancaster
Keyword(s):  
Unit Circle ◽  
Real Line ◽  
Matrix Functions ◽  
Rational Matrix ◽  
The Real ◽  
Minimal Realizations

AbstractA theory of minimal realizations of rational matrix functions W(λ) in the “pencil” form W(λ) = C(λA1 − A2)−1B is developed. In particular, properties of the pencil λA1 − A2 are discussed when W(λ) is hermitian on the real line, and when W(λ) is hermitian on the unit circle.

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.

The Restricted Arc-Width of a Graph

10.37236/1734 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
David Arthur
Keyword(s):  
Unit Circle ◽  
Single Point ◽  
Minimum Width ◽  
Graph Operations ◽  
Function Mapping

An arc-representation of a graph is a function mapping each vertex in the graph to an arc on the unit circle in such a way that adjacent vertices are mapped to intersecting arcs. The width of such a representation is the maximum number of arcs passing through a single point. The arc-width of a graph is defined to be the minimum width over all of its arc-representations. We extend the work of Barát and Hajnal on this subject and develop a generalization we call restricted arc-width. Our main results revolve around using this to bound arc-width from below and to examine the effect of several graph operations on arc-width. In particular, we completely describe the effect of disjoint unions and wedge sums while providing tight bounds on the effect of cones.

scholarly journals Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales

2020 ◽  
Vol 18 (1) ◽  
pp. 353-377 ◽  
Author(s):  
Zhien Li ◽  
Chao Wang
Keyword(s):  
Time Scales ◽  
Quaternion Algebra ◽  
Matrix Exponential ◽  
Dynamic Equations ◽  
Necessary Condition ◽  
Quaternion Matrix ◽  
Exponential Functions ◽  
Cauchy Matrix ◽  
Adjoint Matrix ◽  
Sufficient And Necessary Condition

Abstract In this study, we obtain the scalar and matrix exponential functions through a series of quaternion-valued functions on time scales. A sufficient and necessary condition is established to guarantee that the induced matrix is real-valued for the complex adjoint matrix of a quaternion matrix. Moreover, the Cauchy matrices and Liouville formulas for the quaternion homogeneous and nonhomogeneous impulsive dynamic equations are given and proved. Based on it, the existence, uniqueness, and expressions of their solutions are also obtained, including their scalar and matrix forms. Since the quaternion algebra is noncommutative, many concepts and properties of the non-quaternion impulsive dynamic equations are ineffective, we provide several examples and counterexamples on various time scales to illustrate the effectiveness of our results.

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