The Bounds For The Eigenvalue Of The Special Product Of M-Matrices

M-matrix and non-negative matrix are matrix types with unique properties in computational mathematics,and they are widely used in the economics and other fields.Many algebraists and geometers have studied the eigenvalues of M-matrix and non-negative matrix since the 19 th century.In particular,the problem of the eigenvalue bounds of Hadamard product and Fan product of M-matrix and non-negative matrix has been widely concerned and studied.So far,many results have been achieved for the problem of eigenvalue estimation.This paper will continue to study this kind of problem and give some more accurate and easy-to-calculate new estimates of the upper bounds for the spectral radius of the Hadamard product of non-negative matrix A and the inverse of M-matrix B,the lower bounds of the minimum eigenvalue of the M-matrix B,the lower bounds of the minimum eigenvalue of the Hadamard product of the M-matrix A and its inverse matrix and the minimum eigenvalue lower bound of the Fan product of two M-matrices A and B.The specific content is as follows:Firstly,aiming at the problem of the upper bounds for the spectral radius of the Hadamard product of non-negative matrix A and the inverse of M-matrix B.We derive two inequalities between the elements of the inverse matrix of matrix B,so that the estimation of the upper bounds of spectral radius is closer to the true value by using the eigenvalue inclusion domain theorem and optimally scaled matrix.Then,by using the properties of the non-negative and irreducible M-matrix B,two new estimates about ?B the lower bound are obtained.The numerical results verify their effectiveness and the obtained estimates improve some existing results.Secondly,on the lower bounds of the minimum eigenvalue of the Hadamard product of the M-matrix and its inverse matrix.We derive inequality between the elements of the inverse of matrix,so that the new estimation formulas only depends on the matrix elements by using the correlation properties of the Hadamard product of M-matrix and the eigenvalue existence field theorem of matrix.Numerical experiments show that the new estimation formula is better than the existing estimation formulas.Finally,we use two different eigenvalue existence domain theorems to improve the estimation formulas of the minimum eigenvalue lower bound of the Fan product of two M-matrices A and B,and obtain some new estimation formulas.The results of numerical experiments show that they are effective,and the resulting estimates improve some existing results.There are still many problems worthy of research on the eigenvalue bounds for special products of matrix.At the end of the paper,some ideas for further research are given.