For the problem of the minimum eigenvalue for the Hadamard product on nonsingular Mmatrix and its Inverse ,firstly,recalling the domain theorem of the eigenvalues for the matrix and the estimation formula for the elements of inverse matrix are used in the literature. Secondly,when Ais nonsingular M matrix and A-1 are doubly stochastic,τ(AA－1) is given by combining with the relative properties of the Hadamard product of M matrix and the construction and reduction techniques of inequalities,which is only related to the elements of the matrix,and theoretical analysis proves that it improves the results of existing literature;Finally,numerical examples show that the new lower bound is more accurate than the existing lower bound.