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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. AV BW EVF C, MV NW GVHD ) ,WV 0 V 0 W ) ,WV 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. |