Problems Related To M-matrix And Inverse M-matrix

M-matrix and inverse M-matrix are important special matrices.They are widely used in economics,operations research,engineering and so on.In this thesis,the inverse M-matrix,the generalized inverse of a matrix and the H-matrix are researched by using some equivalence theorems of an M-matrix,and some judgement theorems and equivalence theorems are obtained.The main research work of this thesis is as follows:Firstly,by using the equivalence theorem of complete maximum principle and domination principle and the relationship between an M-matrix and a diagonally dominant matrix,we obtain the necessary and sufficient condition that the transpose of matrix is a potential matrix.Furthermore,based on the positivity of inner product,the equivalent statement of this necessary and sufficient condition is given.Subsequently,based on the conclusion that if a matrix A is an inverse M-matrix,then it is an inverse H-matrix,and with the help of the theorem that A,A-I,I-A^-1 are invertible H-matrices when the comparison matrix(44)(A)of A is a row diagonally dominant matrix,the result that A-I is an inverse M-matrix if and only if A andI-A^-1 are inverse M-matrices is extended to an inverse H-matrix and a singular H-matrix.Secondly,by using the fact that the group inverse of an M-matrix A is nonnegative in the value space R(A),we obtain that the group inverse of the new matrix,formed by the matrix whose group inverse is an M-matrix under rank 1 perturbation,is still an M-matrix.After that,by using the relationship between a block diagonal M-matrix and its sub blocks,we characterize the properties of the original matrix when the group inverse of a block diagonal matrix with special form is an M-matrix.Next,the Moore-Penrose inverse of a matrix is considered.The result that the Moore-Penrose inverse of a matrix is an M-matrix is a Z-matrix under rank 1 perturbation is given,and the existence of product decomposition of the Moore-Penrose inverse of a nonnegative symmetric singular matrix is proved.Finally,by using the basic definition of a row diagonally dominant matrix,we get the conclusion that the left multiplication of a row diagonally dominant inverse M-matrix by a positive diagonal matrix is still a row diagonally dominant inverse M-matrix.On this basis,it is proved that the combination matrix of left multiplication of an inverse M-matrix by a positive diagonal matrix is an M-matrix and the conclusion that the inverse M-matrix is an invertible H-matrix is overturned.Furthermore,a new sufficient condition for a matrix interval to be an inverse M-matrix interval is proved,and some sufficient conditions for a certain matrix interval to be an H-matrix interval are obtained by means of the relationship between the spectral radius of the Jacobi iterative matrix of the H-matrix and its comparison matrix,and the relationship between the H-matrix and its Z-part.