The Study Of Diagonally Dominant Matrix

Generalized strictly diagonally dominant matrices is equivalent to nonsingular H–matrix, it is a kind of matrix which is widely applied in the computational mathematics, it has strong theoretical values and widely actual backgrounds. Therefore, people pay lots of attentions to the study of this kind of matrix. It plays important roles in the fields of information theory,systematic theory,modern economics,control theory and so on. However, it is difficulty to judge the feasibility, so the research and judgment on this kind of matrix attract many scholars. Especially in the 20th century, the research of generalized strictly diagonally dominant matrices is developing quickly, people obtain a lot of beneficial achievements about it.On the basis of previous work, we do further study and improve as well as extend some recent research production.In the first chapter, it chiefly introduces the research background and the practical significances, and then it introduction some symbols and definitions which will be used in this paper.In the second chapter, First, it introduction some lemmas which are used in the paper, and then it lists the main conclusion of this paper, it uses the concepts and properties ofα-diagonally dominant matrices,α-chain diagonally dominant matrices andα-double-stranded diagonally dominant matrices, and then introduction some simple judgment theorems which are used to judge the generalized strictly diagonally dominant matrices, namely nonsingular H–matrix. Finally,the feasibility and superiority of conclusion is proved by the numerical examples. Not only it perfects the generalized strictly diagonally dominant matrices theory and nonsingular H - matrix theory, but also it has laid the solid foundation for the study of computational mathematics, matrix theory, cybernetics and many other fields.