Localization theorems and perturbation analysis on quaternionic eigenvalue problems

Abstract

Quaternions are extensively used in programming video games, computer graphics, control theory and quantum physics, etc. A solution of a quaternionic linear di erential equation with constant coe cients can be presented in terms of right eigenvalues as well as their corresponding eigenvectors of the associated quaternionic matrix. The study of a quaternionic linear di erential equation with constant coe cients is based on nding the zeros of its corresponding quaternionic polynomial. In contrast to the complex case, the location of left eigenvalues of a quaternionic matrix plays an important role in the characterization of zeros of quaternionic polynomials. The stability of linear di erence/di erential equations with quaternionic matrix coe cients is based on the location of right eigenvalues of their corresponding block companion matrices. This thesis mainly deals with localization theorems for the left and right eigenvalues of a quaternionic matrix and their applications for nding bounds/location of zeros of quaternionic polynomials. Bounds for the left and right eigenvalues of quaternionic matrix polynomials are derived. In the proposed research work we also discuss about perturbation bounds for right eigenvalues/generalized right eigenvalues of a quaternionic matrix/quaternionic matrix pencil. The entire work of this thesis is divided into seven chapters and has been brie y described below: Chapter 1 describes preliminaries and basic facts related to the development of our theory. In Chapter 2, inclusion regions for eigenvalues of a quaternionic matrix are derived and bounds for the zeros of quaternionic polynomials are presented. In this chapter, we study Gerschgorin, Ostrowski, and Brauer type theorems for the left and right eigenvalues of a quaternionic matrix. Thereafter a su cient condition for the stability of a continuoustime quaternionic system is given. Chapter 3 presents inclusion regions of zeros of quaternionic polynomials.Chapter 4 discusses basic properties of regular quaternionic matrix pencils, localization theorems of generalized right eigenvalues of quaternionic matrix pencils, and their applications. Chapter 5 derives the de nitions of the left and right eigenvalues of quaternionic matrix polynomials. Next, we present bounds of left and right eigenvalues of quaternionic matrix polynomials. A su cient condition for the stability of a discrete-time quaternionic system is given. Furthermore, bounds for the absolute values of the left and right eigenvalues of quaternionic matrix polynomials are devised and illustrated for the matrix p-norm, where p = 1; 2;1; and F (Frobenius). The above results generalize bounds for the absolute values of the eigenvalues of complex matrix polynomials. Chapter 6 gives the concept of perturbation bounds for right eigenvalues/generalized right eigenvalues of a quaternionic matrix/quaternionic matrix pencil. In particular, Bauer-Fike type theorems for right eigenvalues/generalized right eigenvalues of a diagonalizable quaternionic matrix/diagonalizable quaternionic matrix pencil are derived. Then, a relative perturbation bound for right eigenvalues of an invertible diagonalizable quaternionic matrix is given. Perturbation bounds of right eigenvalues of a quaternionic matrixand perturbation bounds for the zeros of quaternionic polynomials are presented. Finally, in Chapter 7, we give conclusions of our research work and future prospect of this work.