The Diagonally Dominant Degree Of Schur Complement Of Doubly Strictly Diagonally Dominant Matrices And Its Applications

The Schur complement of a nonsingular matrix and its diagonally dominant degree,estimation of the infinite norm of the inverse of a nonsingular matrix,and the eigenvalue location of matrices play an important role in solving linear equations and many fields of linear algebra.In this thesis,we study the three problems for doubly strictly diagonally dominant matrices,and give the upper and lower bounds of the diagonally dominant degree of Schur complement of doubly strictly diagonally dominant matrices.From this,we obtain the eigenvalue localization theorem of Schur complement and the upper bound of the infinite norm of the inverse of the Schur complement.The details are as follows:In chapter 1: We briefly describes the research background and significance of the topic,the research status at home and abroad,and lists the basic concepts,definitions and theorems used in this paper.In chapter 2: By using the technique of inequality expansion and contraction,the upper and lower bounds of diagonally dominant degree of Schur complement of a doubly strictly diagonally dominant matrix are given.It is proved that the new upper and lower bounds are more accurate than the existing results,and are verified by numerical examples.In chapter 3: By using the upper and lower bounds of diagonally dominant degree of Schur complement of strictly doubly diagonally dominant matrix,the more accurate eigenvalue location theorem of Schur complement and the more accurate upper bound of infinite norm of inverse matrix are given,and numerical examples are given to verify the effectiveness of the theoretical results.In chapter 4: We summarize the work done in this paper and put forward the future research problems.