The Diagonally Dominant Degree And Distribution Of Eigenvalues For The Schur Complement Of Some Special Matrices

Many theoretical and practical problems often due to solving of large linearsystems of equations. As one of the most important methods of reducing order,Schur complements are playing a fundamental role in various methods and tech-niques related to the solution of systems of linear equations.In this paper, based on some recent methods, we consider the Gersˇgorindiscs separation from the origin for H-matrices and their Schur complements,showing that the separation of the Schur complement of a H-matrix is greaterthan that of the original grand matrix. Then we reveal the advantages of theSchur-based iteration by a numerical example as application.In chapter one, we introduce some background knowledge of Schur comple-ment, including the meaning and recent works of the topic. Then we present thesummary of this paper, and several basic symbols, definitions as well.In chapter two, by considering the element characteristics of matrix, con-structing positive diagonal matrix, applying some techniques of inequalities andusing the properties of γ-(chain) diagonally dominant matrices, we get γ-chaindiagonally dominant degree, diagonally dominant degree, γ-diagonally dominantdegree for some special matrices and their Schur complements. Our results im-prove and generalize some recent ones.In chapter three, based on the results of chapter two, considering the Gersˇgor-in discs separation from the origin for diagonally dominant matrices and theirSchur complements, we get the localization of eigenvalues and the estimation ofspectral radius for the Schur complement of some special matrices. Our results arebetter than some recent ones. Then, we reveal the advantages of the Schur-basediteration by a numerical example.