Bound Estimations With A Parameter For The Minimum Eigenvalue Of The Hadamard Product Of An M-matrix And Its Inverse

Nonnegative matrix and M-matrix have a wide range of application background which are involved in physics,biology,operations research,finance and other aspects of the study.Hadamard product and Fan product are two special matrix products.In mathematical research,more and more problems will involve the numerical calculation of Hadamard product and Fan product.Moreover,eigenvalue estimation of matrices has always been a hot topic in matrix analysis.In practical application,it is involved in the graph theory method of finding the weight coefficient and the design of production system and other aspects.However,inverse matrix and eigenvalues become more difficult to solve when the dimension of the matrix is very large.Therefore,many scholars devote themselves to estimating the elements of the inverse matrix and the corresponding eigenvalues through the known matrix elements.So the research of bound estimation is of great significance.In this paper,we mainly give several new bound estimations for the eigenvalues of Hadanard product or Fan product of nonnegative matrices and M-matrices.These bound estimations are based on several eigenvalue inclusion sets with a parameter ?(??[0,1]),including four eigenvalue inclusion sets of a diagonally dominant type,? chain diagonally dominant type,double ? diagonally dominant type and double ?chain diagonally dominant type.In previous studies,Gersgorin Circle Theorem and Cassini Oval Theorem were usually used,but the eigenvalue inclusion sets with a parameter were not used to do in-depth research on the estimation of eigenvalues.Therefore,this paper chooses to start from this kind of eigenvalue inclusion sets with a parameter for research.Firstly,through these eigenvalue inclusion sets with a parameter,combined with the related properties of nonnegative matrices and Hadamard product,and using the related techniques of inequality scaling,the upper bound estimations of the spectral radius of the Hadamard product of two nonnegative matrices are obtained.Then,a numerical example is given and appropriate ? is selected to illustrate the superiority of the bound estimation compared with some previous results.Secondly,in combination with the related properties of M-matrices and Fan product as well as the related skills of inequality scaling,the lower bound estimations for the minimum eigenvalue of the Fan product of two M-matrices are obtained through the eigenvalue inclusion sets with a parameter in this paper.A numerical example is also given to illustrate the superiority of the bound estimation by selecting the appropriate ?.Finally,combined with the related properties of M-matrices and Hadamard product as well as the related skills of inequality scaling and the relationship between the elements of inverse matrix and the elements of original matrix in the lemma in this paper,the lower bound estimations for the minimum eigenvalue of the Hadamard product of M-matrix and M-matrix's inverse are obtained by using the eigenvalue inclusion sets with a parameter.Two of the results are compared theoretically.Then,numerical examples are given to illustrate the superiority of the bound estimation over some existing results.Different example picks different ?.Moreover,a large number of 4 dimension M-matrices are constructed randomly by MATLAB for numerical experiments to illustrate the superiority of the bound estimations based on the eigenvalue inclusion sets with a parameter in this paper.