Several Inequalities Of Eigenvalues Of The Special Matrix

The Hadamard product of two nonnegative matrices and the Fan product of two M? matrices are special matrices product. In mathematics, many problems can be transformed into computational problems of the Hadamard product and the Fan product.The upper bounds for the largest eigenvalue ?? BA?? of the Hadamard product of two nonnegative matrices A and B,the lower bounds for the minimum eigenvalue ? BAq ??of the Fan product of two M? matrices A and B and the lower bounds for the minimum eigenvalue? ??1BAq ? of the Hadamard product of A and1B?are research hotspots in matrix. In this paper, by using all simple circuits of the directed digraph of matrix and Cauchy-Schwarz inequality, we will conduct further research. In theory, we prove that the results obtained in this paper are more accurate than the corresponding results. Some results of the comparison are also considered. To illustrate our results, some examples are given.First, let A and B are nonnegative matrices, by using all simple circuits of the directed digraph of matrix and Cauchy-Schwarz inequality,we will give some sharper upper bounds for the largest eigenvalue ?? BA?? of the Hadamard product of A and B.These results are more general, convenient, applicable, essentially improved the related results.Second, if A and B are nonsingular M ? matrices, by using Cassini oval region and Cauchy-Schwarz inequality, we will give some new lower bounds for the minimum eigenvalue ? BAq ?? of the Fan product of matrices A and B. At the same time, we will give some numerical examples to illustrate the efficiency of our results.Finally, let A is an nonsingular M ? matrix, B is an n?n strictly diagonally dominant matrix by row. In this paper, by using the results of relevant literature and Cauchy-Schwarz inequality, we will give more accurate conclusions for the minimum eigenvalue? ??1BAq ? of the Hadamard product of matrices A and1B?.