Estimate To The Minimum Eigenvalues Of M-Tensor

The smallest eigenvalue problem of M-tensors is not only an important branch of Tensor theory, but also have applications in many fields, such as the stationary distribution of high order Markov chain in the statistics, the positive definite of the even-order polynomial in the automatic control system. In this thesis, we focus on the estimate of the smallest eigenvalue problem of M-tensors, and obtain the following results:(Ⅰ) We pointed out an mistake on the estimate of the smallest eigenvalue problem of irreducible M-tensors in [J. He, T.Z. Huang. Inequalities for M-tensors. Journal of Inequalities and Applications,2014,2014(1):1-9], analysis it, and modify it to give a new upper bound and lower bound. Subsequently, the comparisons between the new bounds and the exiting bounds are established, and it is shown that the new bound is better than the existing bound. Numerical examples are given to verify the corresponding results.(Ⅱ) Based o on the characteristic equations of the smallest eigenvalue problem of M-tensors, by the inequality technique, we obtain another upper bound and low bound, and compare the new bound in this section with the obtained results. Numerical examples are given to show that the new bounds are better than the exiting ones.(Ⅲ) by the lower bounds of the smallest eigenvalue problem of M-tensors, some sufficient conditions for the positive definiteness of symmetric tensors, con-sequently, for the positive definiteness of an multivariate polynomial are given.