New Eigenvalue Inequalities for the Hadamard Product and Fan Product of Structured Tensors

The spectral radius of nonnegative tensors and the minimum $H$-eigenvalues of strong $\mathcal{M}$-tensors are two types of tensor eigenvalues with important research significance, which promotes the tensor eigenvalue inequality to become an important component in tensor analysis. In this paper, based on Brualdi-type and Brauer-type eigenvalue inclusion sets of tensors, some Brualdi-type inequalities on the spectral radius for Hadamard product of two nonnegative tensors and some Brauer-type inequalities on the minimum $H$-eigenvalue for the Fan product of two strong $\mathcal{M}$-tensors are provided, respectively. In addition, the theoretical comparisons between the newly obtained inequalities and some previous ones are investigated. Finally, some numerical examples are reported to show the feasibility and effectiveness of our theoretical results.