The Methods For Judging Some Special Types Of Matrices And Their Numerical Algorithms

Generalized diagonally dominant matrix, M-matrix andH-matrix play an important role in numerical algebra,economy,cybernetics theory and so on. Many scholar abroadand home have studied their properties, application and themethods for judging them and obtained a lot of results. Thekey to judge a matrix to be a generalized diagonally dominantmatrix or not is constructing an appropriate positivediagonal matrix. Many scholars got many methods for judgingthem by applying some techniques in matrix theory andinequalities. The thesis considers the methods for judginggeneralized diagonally dominant matrix ,M-matrix which canbe viewed as an extension and improvement of thecorresponding result recently. In chapter one, we discuss generalized block diagonallydominant matrix , by using the techniques of determinantoperation and some skill in inequalities, we construct apositive diagonal matrix and give some necessary conditionsfor judging generalized block diagonally dominant matrix,then we apply the results in point diagonally dominantmatrix and improve the recent results; In chapter two, by using the estimation for the row ofmatrix and improving some previous technique and combiningwith the methods in chapter one, we obtain a set ofconditions for verifying generalized strictly diagonallydominant matrix and get a comparison theorem which suggestthat these theorem improve the latest results; 2In chapter three, firstly, we give a sufficient andnecessary condition for judging generalized diagonallydominant matrix. Secondly, by constructing some differentkinds of positive diagonal matrices D1 and D2, we get somecriteria for judging generalized diagonally dominant matrixwhich improve some previous judging theorem; In chapter four, by utilizing some properties of Schurcomplement of matrix, we give a new necessary and sufficientconditions for judging non-singular M-matrices. We also usethe methods that degrade gradually the rank of the matrixA, and judge a number whether satisfies the conditions, ifthe number satisfies the conditions then A is a nonsingularM-matrix, or A is not a nonsingular M-matrix.