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In this paper, by characterizing the ratio of the smallest and largest values of a Perron vector, we propose some Ostrowski-type bound estimations for the spectral radius of weakly irreducible nonnegative tensors which improve the existing results. Based on Brualdi-type eigenvalue inclusion sets and Gershgorin eigenvalue inclusion sets, we establish some new Ky Fan-type theorems for weakly irreducible nonnegative tensors. [ABSTRACT FROM AUTHOR]

Identifying the positive definiteness of an even-order homogeneous multivariate form is an important task due to its wide applications in such as medical imaging and the stability analysis of nonlinear autonomous systems via Lyapunov’s direct method in automatic control and multivariate network realizability analysis. In this paper, based on the equivalence of the positive definiteness of the form to that of the underlying tensor, and the links between the positive definiteness of a tensor with strong H -tensor, we propose an H -tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms. The validity of the iterative scheme is guaranteed theoretically and the given numerical experiments show the efficiency of the scheme. [ABSTRACT FROM AUTHOR]