Research On M-matrices And Their Related Matrices

Matrix is an important branch of algebra,it not only has rich research value itself,and it is important tool in many branches of mathematics and other natural science.It has important applications in mathematics and other fields.At present,the matrix theory has been widely used in wireless communications,financial statistics,system engineering,optimization theory,the electronic simulation and other fields of engineering.This paper is based on the study of the experts and scholars in the matrix,nonnegative irreducible matrix,and positive definite matrices are studied,some interested conclusions are obtained.The article is divided into four chapters,the main content of the chapters are as follows:The first chapter,we introduce some symbols,basic definitions and basic theorems,including the stable matrix,permutation matrix,nonnegative matrix,at the same time,some known conclusions are introduced.The second chapter,firstly,the basic properties of matrix are used,product of matrix and the convex combination properties is discussed,the information about the product of matrix and the convex combination related conclusions are obtained;By using the properties of the comparing matrix and the nonnegative matrices,and the inverse of the matrix and the properties of the determinant are discussed,matrix inequalities are derived.Secondly,by using some properties of Hadamard product,Hadamard product of the M-matrix is studied,and the important conclusions about the Hadamard product of the M-matrix and related inequalities are obtained;by using the relationship between spectral radius and minimum eigenvalue,the properties of the M-matrix minimum eigenvalue are discussed,the inequalities of the M-matrix minimum eigenvalue are obtained.Finally,by using the basic properties of the M-matrix,the inverse M-matrix,the P-matrix,and D-stable matrix are studied,some important conclusions of the inverse M-matrix,the P-matrix,D-stable matrix are obtained.The third chapter,by using the definition of reducible matrix,the reducible matrix,irreducible matrix and the nonnegative matrix are discussed,the equivalence conditions of the reducible matrix,the linear properties of irreducible matrix and the inequality of nonnegative matrix are obtained.The fourth chapter,by using the basis properties of positive definite matrix and the M-matrix,comparing the positive definite matrix and the M-matrix,the similar conclusion of positive definite matrix and the M-matrix are obtained,and then by using the existing inequalities of positive definite matrix,the inequalities of positive definite are discussed,some important conclusions of positive definite matrix are obtained.