Some Properties Of H-matrices And Block Matrices

H-matrix and Block Matrices play an important role in matrix theory and real applications.It has numerous applications in computational mathematics, matrix theory,numerical algebra,mathematical physics,cybernetics theory,electrical system theory,mathematics of economics,statistics and so on. Many scholars abroad and home have obtained lots of criteria for identifying H-matrix by using iterative arithmetic and techniques in matrix theory and inequalities, and studied its properties and applications.Especially,the theory of generalized H-matrix play an more important role in study of many real question.In this paper,we study the determinat conditions and properties for Hmatrix, give some new criteria for nonsingular H-matrix,properties of the KhatriRao product of block diagonally dominant matrices,the Hadmard product of generalized H-matrices and generalized M-matrices and its application in block iterative methods.In chapter one,we introduce the applied background,present conditions of research and theory and real application on H-matrix.Especially,emphasize the applied background and some related results on generalized H-matrix and block diagonally dominant matrices.In chapter two,N is divided into N₁(?) N₂(+) N₃,combining with the properties of elements in the block diagonally dominant matrix,and using the part elements in the N₁ and N₂ rows,we choose coefficient factors d_i andδ_i that are smaller than 1,and multiply some part elements in the N₁ and N₂ columns by the selected factor,and construct a different positive diagonal matrix D.By inequalities techniques,we obtain some new results.At the same time,the judging methods for the H-matrix with a chain of nonzero elements are presented.Moreover its effectiveness is illustrated by some numerical examples.In chapter three,we main research the block diagonally dominant matrix's Khatri-Rao product under matrix norm and its important function in computational mathematics and statistics.We get that there are several kind of block diagonally dominant matrix's Khatri-Rao product still keep diagonally dominant under some matrix norm,and we also expanded some conclusions recently.In chapter four,The generalized H-matrices has extensive applications in many subjects,such as the convergence of iterative schemes for linear systems arising in the numerical solution of partial differential equations,we prove that the Hadmard product of generalized M-matrices are generalized M-matrices,and the Hadmard product of generalized H-matrices are generalized H-matrices.We also extend its application of generalized iterative methods for linear systems.