Properties, Criteria And Application For Block Diagonally Dominant Matrices

The class of Diagonally dominant matrices is widely concerned by many scientific researchers in many fields,such as numerical algebra,economics,cybernetics and matrix theory itself etc..It is also widely used at engineering technology applications. So researching the class of diagonally dominant matrix's properties is significative to theory research and practical application.Along with the development and the popularization of computer technology,the matrices partitioning technology was widely used in the study of the matrix theory.The block special matrices,especially the block diagonally dominant matrices,plays an important role in many science and engineering calculation problems such as the convergence of block iterative schemes for linear systems from numerical solutions of Euler equations and the study of invariant tori of dynamical system research.Up to now,within the scope of the field,many researchers have acquired some valuable results in many fields,such as properties of block diagonally dominant matrices, decision methods and the convergence block iterative algorithm for it etc..In this paper,we mainly researched the properties,the decision methods and the applications of the class of block diagonally dominant matrices under matrix norm and the class under the positive definite matrix conditions.Our main results are as follows:Firstly,by using the norm propertices of the matrix's Kronecker product we discussed some kinds of diagonally dominant matrices product properties under the matrix norm condition.We have abtained that some kinds of diagonally dominant matrices Kronecker product still keep the diagonally dominant property.Secondly,by using the method of constructing special low-rank matrices,combining with the property of Schur complements and block diagonally dominant matrices,we studied numerical value characteristic,distributing of eigenvalue and estimate of determinant for block diagonally dominant matrices and its Schur complements. We researched that the diagonally dominant properties for the Schur complements ofâ… (â…¡)-block(double) diagonally dominant matrices.We obtain that the block diagonally dominant degree of each row of the Schur complement of anâ… (â…¡)-block(double) diagonally dominant matrix excel the block diagonally dominant degree of relevant row of its original matrix.We use the original matrix to represent the eigenvalue disc separation of the Schur complement ofâ… (â…¡)-block diagonally dominant matrices.For more,we present some upper and lower bounds for the determinants and the distribution of eigenvalues forâ… (â…¡)-block(double) diagonally dominant matrices.Lastly,by using the matrix's continuous transition method,the submatrix's estimate of spectral radius method and some other methods,we studied some decision methods and properties for generalized M-matrix and generalized H-matrix and the applications in block iterative algorithm under positive definite condition. We also discussed the relationship of different block diagonally dominant matrixes. When a block matrix degenerates a point matrix,these decision methods namely become M-matrix's and H-matrix's decision methods.We developed a new way of the decision method of generalized diagonally dominant matrix class.