The New Bounds Of The Eigenvalue Of The Hadamard Product And The Fan Product Of Two Matrices

Many mathematical models based on natural phenomena and man-made laws can be reduced to a partial differential equation.When we need to solve the numerical solutions of partial differential equations,it is often converted into solving the spectral components of a special matrix product or spectral analysis.The Hadamard product and the Fan product are two natural matrix multiplications,which play important role in mathematics and physics.In this paper,we study the upper bound of the spectral radius of the Hadamard product of two nonnegative matrices,the lower bound of the minimum eigenvalue of the Fan product of two M-matrices,and the lower bound of the minimum eigenvalue of the Hadamard product of an M-matrix and inverse M-matrix.At first,due to S-type eigenvalues containing sets,the spectral properties of M-matrices and nonnegative matrices,and inequality scaling,the upper bound of the spectral radius of the Hadamard product of two nonnegative matrices,and the lower bound of the minimum eigenvalue of the Fan product of two M-matrices are obtained.Then their accuracy are verified through concrete examples.On the study of the lower bound of the minimum eigenvalue of the Hadamard product of M-matrix and inverse M-matrix,firstly,we use the lemma to obtain inverse M matrix elements with its inverse matrix elements restrict.Then oval area of the lower bound is presented by S-type eigenvalues.Finally,the minimum value of the region is obtained by using a series of unequal equations,which is the lower bound.In this paper,we present two lower bounds of the minimum eigenvalue of the Hadamard product of an M-matrix and its inverse matrix.The one is the direct inference of the lower bound sequence of the minimum eigenvalue of the Hadamard product of an M-matrix and inverse M-matrix,which is a special case.For the other,the elements of the inverse matrix are limited by the elements of the original matrix.Then the initial lower bound is obtain by matrix inequality and the lemma 2.6.And then,a parameter is introduced and the initial lower bound is approximated by a similar method to get a more accurate lower bound.It is proved that both new lower bounds are more accurate than some of the existing results.Finally,we construct enough M-matrices satisfying the criteria by designing a random algorithm for 4 dimension M-matrices.We carried out a large number of numerical experiments,including the comparison between the two new lower bounds and the existing results and the comparison between the two lower bounds.A large number of experimental data show the accuracy and effectiveness of these two lower bounds.And comparing these two new lower bounds,the conclusion is that these two new lower bounds are mutually advantageous.