Study On Bounds For Eigenvalues Of Hadamard Product And Fan Product Of Matrices

The Hadamard product of matrices and the Fan product of matrices arise in a widevariety of ways, such as trigonometric moments of convolutions of periodic functions,productsof integral equation kernels, the weak minimum principle in partial differentialequations, characteristic functions in probability theory, association schemes in combinatorialtheory, and so on. Motivated by these problems, estimation of bounds for eigenvalueof Hadamard product and Fan product of matrices has been a focus of attention ofmany researchers.In this paper, lower bounds of the minimum m eigenvalueτ( AoA^-1 ) of the Hadamardproduct of anM matrix A and its inverse A^-1 , lower bounds of the minimum eigenvalueτ(BoA^-1) of the Hadamard product of anM matrix B and inverse A^-1 of anotherM matrixA , lower bounds of the minimum eigenvalueτ( AB) of the Fan product of twoM -matrices A and B and upper bounds of the spectral radiusρ(AoB) of the Hadamard productof two nonnegative matrices A and B are investigated. The concrete result in as fol-lows:First, let A and B be nonsingularM matrices, some new lower bounds for the minimumeigenvalueτ(AoA^-1) of the Hadamard product of A and its inverse A^-1 , and somenew lower bounds for the minimum eigenvalueτ(BoA^-1 ) of the Hadamard product ofB and A^-1 are given. We improve the corresponding results of [1] and [2].Second, let A and B be nonsingularM matrices, some new lower bounds for the minimumeigenvalueτ(AB) of the Fan product of A and B are given. It means that we improvethe corresponding results of [3] and [4].Finally, If A and B are nonnegative matrices, some new upper bounds for the spectralradiusρ(AoB)of the Hadamard product of A and B are given. It is to say that we improvethe corresponding results of [3] and [4].