الفهرس 
Chapter 1. General IntroductionOnly 14 pages are availabe for public view
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Abstract
mathematics, engineering and others, many problems can be transformed
into some linear matrix equations. Linear matrix equations have been one of the main
topics in matrix theory and its applications. Matrix equations are algebraic equations
in which the given data and the unknown values are represented by matrices. They
can be found in a variety of computational techniques in control and systems theory
and its applications, as well as in a variety of other engineering and scientific
domains. Different forms of generalized Sylvester equations express a general form
of the linear matrix equations encountered in control theory and its applications. The
problem of finding a solution to several linear matrix equations has received a lot of
attention in the literature.
Centro-symmetric matrices have recently been studied algebraically:
properties such as the existence of the inverse, the expression of the determinant,
and the characterization of Eigen spaces in the case of square matrices have been of
interest. Theoretical results for this class of matrices have applications in a wide
range of statistical fields. Recently, many iterative algorithms are constructed for
solving linear matrix equations over generalized centro-symmetric matrices
The main goal of this thesis, which is divided into five chapters, is to look at
two major issues. The first problem, Problem I, is concerned with introducing an
iterative algorithm to solve generalized coupled Sylvester-transpose matrix
equations over generalized centro-symmetric matrices. Problem 2 is concerned with
solving the generalized Sylvester-conjugate matrix equation over generalized
centro-symmetric and generalized centro-Hermitian matrices using some gradientbased
-iterative algorithms.
In chapter one, we present some fundamental definitions, concepts,
terminologies, lemmas and theorems required to develop this thesis. We also
provide a survey of some well-known matrix equations and some recent techniques
for solving them.
In chapter two, an iterative algorithm for solving the generalized coupled
Sylvester matrix equations over the
generalized centro-symmetric matrices ( is proposed. For any initial
generalized centro-symmetric matrices and , a generalized centro-symmetric
solution ( is obtained within a finite number of iterations in the absence of
round-off errors. Two numerical examples are presented to support the theoretical
results, demonstrating the efficiency and accuracy of the proposed algorithm.
In chapter three, a novel gradient-based iterative method for solving coupled
generalized Sylvester matrix (GSM) equations over generalized centro-symmetric
matrices (GCSMs) is proposed. If the matrix equations investigated here are
compatible with the initial GCSM, a generalized centro-symmetric solution (GCSS)
(may be obtained in a small number of iterations in the absence of round-off errors.
This study’s theoretical findings are supported by numerical examples.
The main goal of chapter four is to develop two relaxed gradient based
iterative (RGI) algorithms that extend the Jacobi and Gauss-Seidel iterations to solve
the generalized Sylvester-conjugate matrix equation over generalized centrosymmetric
and generalized centro-Hermitian matrices. It is demonstrated that for
any initial centro-symmetric and centro-Hermitian matrices, the iterative methods
converge to the centro-symmetric and centro-Hermitian solutions. We present
numerical results that demonstrate the efficacy of the proposed approaches.
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VII
In chapter five, the major target of this chapter is to construct a new iterative
strategy in order to find the solution of the aforementioned matrix equation in chapter
three through generalized centro-symmetric matrices by applying a modified
gradient-based form. The performance of our proposed technique is compared to that
the iterative algorithm of the relaxed gradient-based form over generalized centrosymmetric
matrices. To confirm that our constructed technique is convergent, we set
up some conditions. Finally, we illustrate various numerical tests to ensure the
genuineness of our theoretical results as well as the effectiveness of the suggested
algorithm in order to find the solution of the generalized Sylvester-conjugate matrix
equation.
We emphasize that the MATLAB program was used to validate the proposed
algorithms in this thesis, and that the numerical results obtained in the three chapters
of this thesis demonstrate that the proposed algorithms are efficient and accurate.