An Estimation Formula On Lower Bound For The Minimum Eigenvalue Of The Hadamard Product Of Nonsingular Irreducible M Matrix

M matrix is the main branch of computational mathematics,which is widely used to solve the problems about physics,economics and biology etc.And the estimation of lower bounds on the minimum eigenvalue of M matrix is one of the main concepts in the matrix theory,it has important research significance.In this paper based on the existing literature,the lower bound estimation for the minimum eigenvalue of the Hadamard product of a nonsingular irreducible M matrix A and a doubly stochastic matrix-1A is given by using Gersgorin disc theorem,and the lower bound estimation for the minimum eigenvalue of the Hadamard product of two nonsingular irreducible M matrices A and B is given by using the matrix eigenvalue existence theorem.The structure of this paper is organized as follows:In chapter 1,production and the application background and research status of nonnegative matrix,M matrix and the Hadamard product of matrix at home and abroad are given.The research results of this paper are introduced.In chapter 2,first,it is introduced the basic knowledge of the nonnegative matrix,irreducible matrix,M matrix,Hadamard product,spectral radius and so on.Then,it is also introduced some exisiting conclusions,lemmas and theorems.In chapter 3,one of the main achievements of this paper is that the nonsingular irreducible M matrix A and doubly stochastic matrix A^-1 is studied,and by using the Gersgorin disc theorem,the lower bound estimations of the minimum eigenvalue??? of Hadamard product AoA^-1 and proofs are given.The formulas are as follows:andAnd it is proved that these estimations are better than the existing literature,and through numerical examples show that these estimations are more accurate than the estimations of the existing literature,and these estimations are relatively easy to calculate because they only depend on the matrix elements.In chapter 4,another major achievement of this paper is that,on the basis of the existing literature,the lower bound of the minimum eigenvalue of the Hadamard product of a nonsingular irreducible M matrix A and another nonsingular irreducible M matrix B and proofs is given.The formula is as follows:The numerical result shows that the proposed estimator is more accurate than the existing literature.In chapter 5,summary and prospect.The research work of this paper is summarized,and the inadequacies of the paper and the direction of further research are put forward.