Doubly Diagonally Dominant Matrix And Its Promotion, And M-matrix Study Of The Relationship

M-matrices are an important matrix species of the research for numeric algebra-an important branch of computation mathematics. The research for M-matrices plays an important role in computational mathematics and other lots of applied fields.In this paper, we put emphasis on further research for the concepts judgment and equivalent typicality of the two important matrix species, putting forward the new idea about generalized diagonally dominant matrix in order to judge whether the non-doubly diagonally dominant matrix belongs to non-singular M-matrices. And meantime a-doubly diagonally dominant matrices was studied in detail. The paper points out that as long as we examine the related quantity involved in the circuit of the associated digraph of matrices we may determine whether a matrix is a non-singular M-matrices or not.There are three chapters in the paper.In chapter 1, the paper supply us with some basic concepts and related results of matrices theory which was included in the paper, and mainly contains irreducible, weakly irreducible and their property, generalized diagonally dominant matrices * M-matrices and their property.In chapter 2, a subclass of doubly diagonally dominant matrices-irreducibly doubly diagonally dominant matrices was studied. By means of the associated digraph of matrices we obtain the necessary and sufficient conditions of whether irreducibly doubly diagonally dominant matrices belongs to non-singular M-matrices. At the end of the paper, a new type of diagonally dominant matrices-conjugationaly doubly diagonally dominant matrices was put forward, the results obtained needn't matrix to be doubly diagonally dominant in advance, which suits for a widely matrices type.Based on the results, chapter 3 obtains the necessary and sufficientconditions of the positive line a -doubly diagonally dominant matrices being non-singular M-matrices, by means of the nature of the associated digraph of matrices irreducible and weakly irreducible matrices. The results obtained simplify the process of judgment, only making us to check the related quantity involved in the circuit of the associate digraph of matrices. In the end, we make use of the results obtained and combine the method of theory graph to examine several examples.